Microscope generating a three-dimensional representation of an object and images generated by such a microscope

ABSTRACT

A microscope is disclosed which determines a complex three-dimensional representation of an object based on a series of recordings of the light wave diffracted by the object, wherein the direction of the wave lighting of the object varies between two succesive recordings. The diffracted wave interferes with a reference wave on a receiving surface and a frequency representation of the diffracted wave is computed from interference patterns received on the receiving surface. A plurality of frequency representations of diffracted waves are then superimposed yielding a frequency representation of the object. The phase of each frequency representation of a diffracted wave is shifted in order to compensate for variations of the phase difference between the reference wave and the wave lighting of the object.

1. BACKGROUND OF THE INVENTION

This invention relates to a microscope generating a three-dimensionalrepresentation of the observed object, operating on a principle derivedfrom “image formation by scattered field inversion”, tomography andsynthetic aperture systems.

2. THE PRIOR ART

2.1. References

[Wolf]: Three-Dimensional Structure Determination of Semi-TransparentObject from Holographic Data, Emil Wolf, Optics Communications, Vol. 1,No. 4, p.153, October 1969.

[Dändliker]: Reconstruction of the Three-Dimensional-Refractive Indexfrom Scattered Waves, R. Dändliker, K. Weiss, Optics Communications,Vol. 1, No. p.323, February 1970.

[Fercher]: Image Formation by Inversion of Scattered Field Data:Experiments and Computational Simulation. A. F. Fercher, H. Bartelt, H.Becker, E. Wiltschko, Applied Optics, Vol.18, No 14, p.2427, July 1979

[Kawata]: Optical Microscope Tomography. I. Support Constraint, S.Kawata, O. Nakamnura & S. Minami, Journal of the Optical Society ofAmerica A, Vol. 4, No.1, p.292, January 1987

[Noda]: Three-Dimensional Phase-Contrast Imaging by aComputed-Tomography Microscope, Tomoya Noda, Satoshi Kawata & ShigeoMinami, Applied Optics, Vol. 31, No. 5, p.670, Feb. 10, 1992

[Devaney]: The Coherent Optical Tomographic Microscope, A. J. Devaneyand A. Schatzberg, SPIE, Vol.1767, p.62, 1992

[Wedberg]: Experimental Simulation of the Quantitative ImagingProperties of Optical Diffraction Tomography, Torolf A. Wedberg andJacob J. Stamnes, Journal of the Optical Society of America A, Vol. 12,No 3, p.493, March 1995.

[Vishnyakov]: Interferometric Computed-Microtomography of 3D PhaseObjects, Gennady N. Vishnyakov & Gennady G. Levin, SPIE Proceedings,Vol.2984, p.64, 1997

[Ausherman]: Developments in Radar Imaging, D. A. Ausherman, A. Kozma,J. L. Walker, H. M. Jones, E. C. Poggio, IEEE Transactions on Aerospaceand Electronic Systems, Vol. 20, No. 4, p.363, July 1984.

[Goodman]: Synthetic Aperture Optics, Progress in Optics, Vol. VIII,1970, North Holland Publishing Company.

[Walker]: Range-Doppler Imaging of Rotating Objects, Jack L. Walker,IEEE transactions on Aerospace and Electronic Systems, Vol. 16, No. 1,p.23, Januasy 1980.

[Brown]: Walker model for Radar Sensing of Rigid Target Fields, WilliamM. Brown, IEEE Transactions on Aerospace and Electronic Systems, Vol.16, No 1, p.104, January 1980.

[Turpin 1]: U.S. Pat. No. 5,384,573

[Turpin 2]: Theory of the Synthetic Aperture Microscope, Terry Turpin,Leslie Gesell, Jeffrey Lapides, Craig Price, SPIE Proceedings, Vol.2566, p.230, 1995

[Turpin 3]: The Synthetic Aperture Microscope, Experimental results, P.Woodford, T. Turpin, M. Rubin, J. Lapides, C. Price, SPIE Proceedings,Vol. 2751 p.230, 1996

[Lauer 1]: Patent WO 98/13715

2.2. Description of the Prior Art

A three-dimensional object may be characterized optically by a certainnumber of local parameters, for example its index and its absorptivityat each point. Mathematically, this may be expressed by the data at eachpoint of a complex number which is a function of the local parameters atthe considered point. A three-dimensional spatial representation of theobject can then be expressed in the form of a three-dimensional array ofcomplex numbers.

By carrying out the three-dimensional Fourier transform of thisthree-dimensional spatial representation, a three-dimensional frequencyrepresentation of the object is obtained.

[Wolf] showed that a three-dimensional representation of a weaklydiffracting object can be obtained from the acquisition of the wavediffracted by this object when it is illuminated successively by aseries of plane waves of variable direction. [Wolf] also determined themaximum resolution thus obtainable, expressed as a function of theilluminating wavelength. This resolution corresponds to a maximum periodof λ/2 for the sinusoidal components the object's representation, i.e. asampling period in the Nyquist sense of λ/4 which is a resolution twiceas fine as that of conventional microscopes. [Dändliker] improved[Wolf]'s formalism and provided a geometrical interpretation of it. Fromthe wave diffracted by the object under a given illumination, part ofthe three-dimensional frequency representation of the object isobtained. This part is a sphere in a three-dimensional frequency space.By combining the spheres thus obtained for various illuminating waves,the frequency space can be filled, obtaining the three-dimensionalfrequency representation of the object. The latter can then be inversetransformed to obtain a spatial representation.

[Fercher] designed a microscope constituting the first practicalapplication of the principles defined by [Wolf] and [Dandliker]. In thatmicroscope, the wave diffracted by the object is picked up on areceiving surface on which it interferes with a reference wave nothaving passed through the object and the phase of which can be modified.From several interference figures differing from each other by the phaseof the reference wave, [Fercher] obtains, at each point of the receivingsurface, the amplitude and the phase of the wave diffracted by theobject.

[Fercher] does not use several successive illumination waves but severalilluminating waves generated simultaneously by means of a diffractiongrating, thus limiting the number of possible illumination directions,even though the use of several successive illuminating waves does notpresent any particular technical difficulty. The reason for this choiceis not clearly explained. However, it appears that this technique isadopted in order to obtain illuminating waves all having the same phaseat a given point of the image. In fact, the equation (1) of the document[Wolf] assumes that each illuminating wave has a zero phase at the pointof origin of the position vectors.

The method defined by [Wolf], [Dändliker] and [Fercher] is generallycalled “image formation by scattered field inversion”. Anotherconventional approach for obtaining three-dimensional images istomography. Tomography, used for example in x-ray techniques, consistsin reconstructing an image from a set of projections of this image alongdifferent directions. Each projection depends linearly upon athree-dimensional density function characterizing the object. From asufficient number of projections it is possible to reconstitute theobject by reversing this linear correspondence.

Tomography was adapted to optical microscopy by [Kawata]. In histomographic microscope, a plane and non-coherent illuminating wave ofvariable direction is used. This illuminating wave passes through asample and then a microscope objective focussed in the plane of thesample. It is received on a receiving surface placed in the plane inwhich the objective forms the image of the sample. Because theillumination is non-coherent, the intensities coming from each point ofthe object are added and the image intensity produced on the receivingsurface consequently depends linearly on the three-dimensional densityfunction characterizing the absorptivity of the object. From asufficient number of images it is possible to reconstitute the image byreversing this linear correspondence. This microscope differs from usualtomographic systems in that the linear correspondence between thedensity function of the object and a given image is not a projection,but is characterized by a three-dimensional optical transfer function.

This microscope is not very suitable for obtaining images that take intoaccount the index of the sample. [Noda] designed a modified microscopeenabling this phase to be taken into account. The initial idea of thatmicroscope [Noda] is to use phase contrast for obtaining an image whichdepends on the index of the sample, and to adapt to this configurationthe linear correspondence reversal principle already implemented by[Kawata]. The use of the [Noda] microscope is however limited to thestudy of non-absorbing objects whose index variations are extremelysmall.

[Noda]'s text does not refer to holography nor to “image formation byscattered field inversion,” but the operating principle involved can beinterpreted within this framework. In fact, the technique adopted by[Noda] is tantamount to using on the receiving surface a reference waveconsisting of the illuminating wave alone. From the images received fora set of illuminating waves of variable direction, a three-dimensionalfrequency representation is obtained. The complex wave detected on thereceiving surface is replaced here by a pure imaginary value obtained bymultiplying by j the real value obtained using the single reference waveconstituted by the illuminating wave shifted π/2 in phase. If thereference wave is sufficiently higher, at each point of the receivingsurface, than the diffracted wave, then the quantity thus obtained isthe imaginary part of the complex wave really received on the receivingsurface, the phase reference being the phase of the illuminating wave.The object generating a pure imaginary wave equivalent to that detectedby [Noda] on the receiving surface is made up of the superposition ofthe observed real object and a virtual object whose complex spatialrepresentation is obtained from that of the real object by symmetry inrelation to the plane of the object corresponding to the receivingsurface, and by reversing the sign of the real part. Using the imaginarypart thus detected in a manner similar to that used by [Fercher] for thedetected complex wave, a function representing the superposition of thereal object and the virtual object is generated in frequencyrepresentation. During each acquisition, the two-dimensional frequencyrepresentation obtained by performing the Fourier transform of the valuedetected on the receiving surface includes a part corresponding to thereal object and a part corresponding to the virtual object, whichcoincide only at the point corresponding to the illumination frequency.It is thus possible to select only the part corresponding to the realobject, so as to obtain a representation thereof. [Noda] in fact usesthe superposition of the real object with the virtual object which hesymmetrizes in relation to the plane of the object corresponding to thereceiving surface, thus obtaining a pure imaginary representationcorresponding to the imaginary part of the representation that would beobtained using [Wolf]'s method.

The theoretical explanations given in the document [Noda] are verydifferent from those presented here and are perfectly valid. Theprinciple consisting in reversing a filter by multiplication in thefrequency domain, as applied by [Noda], is found to be equivalent to theexplanations given above, although obtained by different reasoning. Itmay be considered that FIGS. 2 and 3 of the document [Noda] illustratehow the three-dimensional frequency representation of the object isgenerated from two-dimensional frequency representations.

[Devaney] proposed a tomographic microscope whose operating mode isderived essentially from the method of [Wolf]. In the [Devaney]microscope the reference wave coincides with the illuminating wave.Consequently, that microscope does not have means for varying the phaseof the reference wave. As in the case of [Noda], the detected waveconsequently corresponds to that which would be formed by thesuperposition of a real object and a virtual object. [Devaney] solvesthe problem by placing the receiving surface outside of the object, sothat the real object and the virtual object do not overlap. When thedirection of the illuminating wave varies, only one of the two objectsis reconstituted. Two variants of the microscope are presented: a firstin which the object is fixed and the direction of the illuminating waveis variable, and a second in which the object rotates around a fixedpoint, the illuminating wave then having a fixed direction in relationto the receiver. The first version of the [Devaney] microscope was builtby [Wedberg].

Another approach allowing the adaptation of tomography to the productionof phase images is that of [Vishnyakov]. [Vishnyakov] introduces areference wave distinct from the illuminating wave and carries out adetection of the wave received on a receiving surface in accordance witha method similar to the one used by [Fercher]. He then generates aprofile characteristic of the phase difference between the received waveand the illuminating wave. As this phase difference is considered to bethe projection of the index along the direction of the illuminatingwave, it regenerates the distribution of the index in the objectaccording to the tomographic method used conventionally in x-raytechniques. This method may be compared to a method of the [Wolf] type,but in which the sphere portion acquired in the frequency space would beassimilated with a plane portion, which is largely unjustified in thecase of an objective with a large aperture such as the one used here.

The technique of the synthetic aperture radar is an imagery method usedin the field of radar waves and which was considered very early forapplication to the field of optics. [Ausherman] presents the history ofthis technique. The application of the synthetic aperture radartechnique to the field of optical wavelengths would make it possible inprinciple to obtain images of an observed object. However, in order forthe technique to be feasible, it is necessary, at all times, to have theposition values, within a reference coordinate system linked to theobject, of each element of the transmitter-receiver assembly. Thesevalues must be known to within a fraction of a wavelength. This isachievable in the area of radar frequencies, in which wavelengths aremacroscopic and can be, for example, of a few tens of centimeters. Inthe optics field, where wavelengths are sub-micrometric, this isdifficult to achieve. This problem is the basic reason for which thesystem is difficult to adapt to optics, as indicated in [Goodman], Pages36 to 39.

[Walker] and [Brown] formalized the synthetic aperture radar method in aform similar to that already obtained by [Wolf] for optical systems.This formalism was originally used by [Walker] with a radar imagerymethod in which the transmitter-receiver assembly is fixed and in whichthe object rotates around a fixed point. This overcomes the problem ofdetermining the position of the object at each instant.

[Turpin] recently described several microscopes constituting anadaptation of synthetic aperture radar principles to the field ofoptics.

In the microscope implemented by [Turpin 3], the material configurationused complies with the principle used in [Walker] to overcome theproblem consisting in determining the position of the object at eachinstant, i.e. the transmitter and receiver are fixed and the objectrotates around a fixed axis. This microscope is also similar to thesecond version of the [Devaney] microscope. Since the rotation axis, thetransmitter and the receiver are fixed, the position of thetransmitter-receiver assembly within a reference coordinate systemlinked to the object can be known with the required accuracy.

However, effective resolution calls not only for a mechanical systemmaking it possible to determine the movement of the object, but also thetaking of this movement into account in the definition of algorithmsand/or appropriate adjustment of the system. In the absence of specialprecautions, the point of origin of the reference wave moves in relationto the object over a circle centered on the rotation axis of the object.If this movement is significant, this effect destroys the image. If thismovement is small, resolution in the plane of this circle is affected inproportion to the amplitude of the movement.

To solve this problem in the absence of any specific compensationalgorithm, the point of origin of the reference wave should be on therotation axis of the object. This condition is in principle difficult toobtain. [Turpin] does not mention this problem and does not specify anymeans of appropriate adjustment.

This problem can however be solved when use is made, for example, of aflat specimen, by performing an adjustment intended to comply with thefollowing conditions:

(i)—The image of the reference wave on the “image CCD array” of FIG. 1in document [Turpin 3] must be a point image.

(ii)—When the object makes a rotation of 180 degrees, the image obtainedmust be symmetrized in relation to an axis passing through the pointimage of the reference wave.

The position of the “image CCD array” must be adjusted to comply with(i).

The position of the entire receiver must be adjusted to comply with(ii).

This solution is however not perfect, depending essentially on a visualinterpretation. It can be used reasonably only for very simple objects.

The microscope described in [Turpin 3] is a particular case of thegeneralized system described in [Turpin 1] and [Turpin 2]. Thegeneralized system specifies that the illuminating wave and/or theposition of the receiver can vary. However, the material configurationsproposed do not allow the solution of the problem consisting indetermining, to within a fraction of a wavelength, the position of thetransmitter and the receiver in relation to the object. In fact, theilluminating wave of variable direction is produced by mechanicaldevices which cannot be controlled with sub-micrometric accuracy.

The microscope described in [Lauer 1] allows the generation of thefrequency representation of a wave coming from the object and thereconstitution of the object from several of these representations. Themethod used in [Lauer 1] is not directly related to the one described by[Wolf]. In fact, in the case in which it recombines several frequencyrepresentations to obtain the representation of the object:

it uses spatially incoherent lighting and not plane illuminating waves

it combines the frequency representations of the waves received byintensity summing in the spatial domain.

The image obtained by [Lauer 1] is affected by a residual granularityeffect, does not allow a differentiation of the refractive index and ofabsorptivity, and does not yield the theoretical accuracy indicated by[Wolf].

3. SUMMARY OF THE INVENTION

3.1. Problem to Be Solved by the Invention

Systems based on “image formation by scattered field inversion,”“tomographic” systems or “synthetic aperture” systems appear to beequivalent to each other, at least when examined from only the coherentoptical domain. Two classes of systems may be distinguished depending onthe reference wave generation mode:

the microscopes of [Noda] and [Devaney] use a reference wave coincidingwith the illuminating wave.

the microscopes of [Fercher], [Vishnyakov] and [Turpin] use a referencewave distinct from the illuminating wave.

Microscopes of the first category exhibit limitations in terms ofobserved image size, characteristics imposed on the object, ordisplayable parameters. These limitations are due to the fact that, inthe absence of a reference wave distinct from the illuminating wave, itis not possible to acquire under good conditions the complex value ofthe wave received on the receiving surface. It is thus necessary to usevarious expedients to eliminate spurious images and various disturbancesgenerated by the drawbacks of the acquisition method.

Microscopes of the second group allow this problem to be solved. Inprinciple, microscopes of the second group should allow therepresentation of the refractive index and the absorptivity of theobject in three dimensions with quarter-wavelength accuracy, accordingto the theory devised by [Wolf] and [Dändliker]. Such performance isclearly higher than that of all existing optical microscopes includingthe confocal microscope, and these microscopes should in all logic haveled to industrial applications. However, none of these microscopes hasas yet made it possible to obtain quality images comparable to thoseproduced, for example, by the [Noda] microscope, and thus thesemicroscopes have not gone beyond the experimental stage. The reason forwhich these microscopes have never yielded high quality images, despitetheir theoretical possibilities, has never been clearly determined.

A first approach to the problem is contained implicitly in the equation(1) of the document [Wolf]: all the illuminating waves must have thesame phase at the origin of the representation. However, in an opticalsystem, the only values accessible to measurement are the phasedifferences between a reference wave and a wave to be analyzed. The factthat at the illuminating waves all have the same phase at a given pointof the object, as indicated implicitly by the equation (1) in thedocument [Wolf], is thus not sufficient to ensure the proper functioningof the system: it is also necessary for the reference wave to meetappropriate conditions, so that the phase differences accessible tomeasurement lead to correct results.

A second approach to the problem is provided by [Goodman], Pages 36 to39, in the terms of the numerical aperture radars: the position of thetransmitter and receiver must be determined with an accuracy within thewavelength, and this is not feasible in the optical domain.

If the illuminating wave has a variable direction, these two approachesare similar: in fact, an indetermination on the position of thetransmitter results, among other effects, in a phase shift of theilluminating wave at the origin of the representation. We shall confineourselves here to these systems, i.e. to the microscopes of [ercher] and[Vishryakov], and to the versions of the microscope of [Turpin] whichinclude a variable direction illuminating wave.

For example, in the microscopes of [Turpin], the illuminating wave isgenerated by a mechanical device. This device does not allow theilluminating wave phase to be controlled. When two successivethree-dimensional images are taken, a given illuminating wavecharacterized by its direction will not have the same phase differencewith the reference wave in each case. The wave detected on the receivingsurface will consequently not have the same phase either, and hencefinally that two successive three-dimensional images taken will not leadto the same result, even in the absence of any noise. This examplehighlights the basic problem which has up to now limited the performanceof microscopes of the second group: lack of control of the phasedifference between the illuminating wave and the reference wave leads tonon-reproducibility of the results obtained and, in general, resultswhich do not correspond to what is expected considering the theoreticalapproach of [Wolf].

3.2. Solution of Problem According to the Invention

The entire three-dimensional frequency representation of an object canbe multiplied by a complex number Ae^(jα). It will then be said that thefrequency representation of the object is affected by a phase shift αand by a multiplicative factor A. If an object is characterized by itsthree-dimensional frequency representation, its spatial representationcan be obtained by taking the inverse Fourier transform of thisfrequency representation. If the frequency representation of the objectis affected by a phase shift α and a multiplicative factor A, itsspatial representation is affected by the same phase shift and the samemultiplicative factor, corresponding to a modification of the functiongiving the complex number associated with a point as a function of thelocal parameters at this point.

It is also possible to multiply each point of the three-dimensionalrepresentation by a complex number Ae^(jα) depending on the point. Twodifferent points of the three-dimensional frequency representation canthen be affected by a phase shift and a multiplicative factor which aredifferent. By performing the inverse Fourier transform of a frequencyrepresentation in which the phase shift and/or the multiplicative factordepend on the considered point, a modified representation of the objectis obtained in which the complex number associated with a point dependsnot only on the local parameters at the considered point, but also onthe local parameters at a set of other points. This modifiedrepresentation of the object is a filtered spatial representation, thefilter having a frequency representation consisting of the numbersAe^(jα) defined at each point. Depending on the characteristics of thisfilter, more or less correct frequency representations will be obtained.

In the microscopes of [Turpin], the illuminating wave is generated by amechanical device. With each change of direction of the illuminatingwave, a random phase shift of this wave occurs, and hence a random phaseshift in the corresponding part of the three-dimensional frequencyrepresentation.

In the document of [Vishnyakov], and for the same reasons, the phase ofthe illuminating wave varies randomly with each change of illuminationdirection. Equation (2) on Page 67 of the document [Vishnyakov] shouldbe replaced by ψ(x,y)=Φ(x,y)+x sin α+φ, where φ is the phase of theilluminating wave at the origin of the three-dimensional representationobtained. When the direction of the illuminating wave varies, the valueof φ varies. Non-determination of the correct value of φ results in theaddition of a constant to each projection obtained, this constantvarying randomly between two projections. This consequently makesinexact the assimilation of the phase profile obtained with theprojection of the index. The method of [Vishnyakov] is roughlyequivalent to the method of [Wolf] in which a portion of the sphere ofthe frequency space, obtained from a given illuminating wave, would havebeen assimilated to a plane. The non-determination of φ is equivalent toa random phase shift of the entire part of the two-dimensional frequencyrepresentation generated from a given illuminating wave. However, othersources of error are added to this effect, and in particular the factthat a conventional tomographic reconstruction is used.

In the document of [Fercher], owing to the use of a diffraction gratingto generate the three illuminating waves simultaneously, there is norandom phase shift in the illuminating waves. However, a detailedanalysis of the system shows that, to obtain the same phase shift in theparts of the frequency representation obtained from each illuminatingwave, the virtual image, in the object, of the point of focus of thereference wave, must coincide with a point in which the illuminatingwaves all have the same phase. This calls for very precise adjustment ofthe position of the reference wave origin. As the document of [Fercher]does not contain any mention of such an adjustment, it is likely that ithas not been carried out. In any case, the solution adopted by [Fercher]limits significantly the number of illuminating waves that may be usedand the aperture under which the wave coming from the object can beacquired.

In existing microscopes using variable direction illuminating waves anda reference wave distinct from the illuminating wave, the parts of thefrequency representation obtained from different illuminating waves arethus affected by different phase shifts. Consequently, the inversion ofthe frequency representation obtained generates a filteredrepresentation which is in general of rather poor quality and which ismoreover not reproducible owing to the random nature of the phase shift.

The invention consists in providing a microscope in which the directionof the illuminating wave is variable, but comprising means forgenerating a three-dimensional representation of the object in which thedistribution of the phase shift affecting each point obtained, infrequency representation, is concentrated around a constant value.Ideally, this phase shift should be constant, but the existence ofdisturbances such as Gaussian noise, a residual spherical aberration, ora small imprecision in controlling the phase difference between thereference wave and the illuminating wave, creates a certain spread ofthe distribution around the constant value.

In existing microscopes using variable direction illuminating waves anda reference wave distinct from the illuminating wave, and in the case inwhich a large number of distinct illuminating waves are used, such as in[Turpin], this phase shift tends to be of a random nature and itsdistribution is hence roughly homogeneous over the interval [0,2π] .Where a limited number of illuminating waves are used, as in [Fercher],the phase shift distribution exhibits peaks of comparable level centeredon several distinct values corresponding to the phase shifts affectingthe sub-representations obtained from each illuminating wave.

The fact that the phase shift affecting each point, in frequencyrepresentation, is roughly constant, constitutes a new functionality ofthe microscope which allows, for example, a better quality spatialrepresentation. However, the generation of a three-dimensionalrepresentation of the observed object does not necessarily constitutethe final purpose sought by the microscope user. For example, themicroscope may be used to read three-dimensional optical memories. Inthis case, the data can be encoded before being stored in the opticalmemory. The microscope then allows a three-dimensional representation ofthis optical memory to be obtained from which the data can be decoded.The three-dimensional representation of the object is then a calculationintermediary finally enabling the decoded data to be obtained.

3.3. Vocabulary Used and General Considerations

A three-dimensional object can be characterized optically by a certainnumber of local parameters. These parameters can, for example, be itsrefractive index and its absorptivity at each point, or its absorptivityand one of its indices in the case of a non-isotropic material. It ispossible to define at each point a complex number which is a function ofcertain of these local parameters, this function generally being defineduniquely in the entire spatial representation and hence not depending onthe considered point. It will then be possible to express athree-dimensional spatial representation of the object in the form of athree-dimensional array of complex numbers. The dependency between thecomplex number and the local parameters can be defined in various ways.For example, this complex number may reduce to a real numbercharacterizing the index, as in the microscope of [Noda]. The definitionthat will be used most often will however be of the type given by[Wolf], but with a “complex index” representing both the refractiveindex and absorptivity. For example, the real part of the complex numbermay be proportional to absorptivity and its imaginary part to therefractive index. A complex number may be used which is obtained byrotating the preceding in the complex plane, corresponding to a phaseshift. In every case, the three-dimensional spatial representation ofthe object is unique if the correspondence between the complex numberand the local parameters has been defined and the central part of therepresentation has also been defined.

Performing the three-dimensional Fourier transform of thisthree-dimensional spatial representation, a three-dimensional frequencyrepresentation of the object is obtained.

The phase shift affecting a point of the three-dimensional frequencyrepresentation of the object is defined when the representation of theobject has been defined uniquely, i.e. when the complex function of thelocal parameters and the point of origin characterizing the spatialrepresentation have been defined.

When these parameters are not specified, it may be considered that thephase shift is defined in relation to the spatial representation whichcoincides at best with the obtained three-dimensional representation ofthe object.

The term “three-dimensional representation of an object” will designateall the numerical data characterizing the spatial or frequencyrepresentation of the object, independent of the manner in which thesedata are combined or stored, for example in a computer memory. Thisrepresentation may be expressed, for example:

in the spatial domain, in the form of a complex number dependent onspatial coordinates

in the frequency domain, in the form of a complex number dependent onthe spatial frequency

in any other manner when the “three-dimensional representation of theobject” allows the spatial or frequency representation of the object tobe obtained by a known algorithm.

Part of the three-dimensional frequency representation of the objectwill be called frequency sub-representation of the object, and the termsub-representation will designate all the corresponding data,independent of how they are combined or stored.

A wave reaching a receiving surface is entirely characterized by itsamplitude and its phase along each polarization direction and at everypoint of the receiving surface. A scalar representation of the wave mayalso be adopted by limiting oneself, for example, to a singlepolarization direction, the wave then being characterized by a singlephase and a single intensity at every point of the receiving surface.From the wave measured on a receiving surface, a frequencysub-representation of the observed object can be generated. Thissub-representation is two-dimensional in that it constitutes a sphereportion in the space of the frequencies, as indicated by [Dändliker]. Acalculation intermediary may be constituted by a frequencyrepresentation of the wave, defined by the wave phase and intensity oneach wave vector, a scalar representation being adopted.

The frequency representation of the wave is two-dimensional and can beprojected on a plane without any loss of information. Such a projectionyields a plane image that will be called “frequency plane image”. In asystem such as that of [Turpin] or that of [Fercher], such a frequencyplane image is obtained directly on the receiving surface. In othersystems, such as the second embodiment of the present invention, such afrequency plane image is obtained by two-dimensional Fouriertransformation of the scalar representation obtained directly on thereceiving surface. A modified frequency plane image can also be obtainedfrom several frequency plane images differing from each other in thepolarization of the illuminating wave and the direction of analysis ofthe wave received on the receiving surface. The frequency plane imagecan constitute a calculation intermediary making it possible to obtainthe frequency representation of the wave and then a correspondingsub-representation of the object.

The term “two-dimensional frequency representation” will designateeither a two-dimensional frequency representation or a two-dimensionalpart of a three-dimensional frequency representation. In particular, itcan designate either a frequency plane image, a frequency representationof a wave, or a two-dimensional sub-representation of the object.

The term “lens” will designate, throughout the text, either simplelenses, compound lenses, or achromats, generally designed to limitoptical aberration.

In the rest of the text, five embodiments are described, referred to asEmbodiments 1, 2, 3, 4 and 5.

3.4. Obtaining the Three-Dimensional Frequency Representation Directly

For each direction of the illuminating wave, a frequencysub-representation of the object is obtained by direct application ofthe methods defined in [Fercher] and [Turpin]. In the systems of[Fercher] and [Turpin], the microscope is built so that the differentsub-representations obtained are affected by different phase shifts.According to one embodiment of the invention, the microscope is built sothat these phase shifts are constant. This variant of the inventionimplies:

(i)—that the microscope is built so that the phase difference between anilluminating wave and the reference wave with which it interferes isreproducible. This condition implies an appropriate construction of themicroscope. This is shown, in the absence of vibrations, in Embodiments3, 4 and 5. Embodiments 1 and 2 are affected by the same problems as themicroscope of [Turpin]: the phase difference between reference wave andilluminating wave varies randomly owing to the overall mechanicaldesign. Embodiments 1 and 2 consequently do not allow compliance withthis first condition. Embodiments 3 to 5 enable this condition to becomplied with because of the different design of the illuminating andreference wave generating system.

(ii)—that the microscope is built so that, through appropriateadjustment, the phase difference between the illuminating wave and thereference wave can be made constant. Embodiment 3 does not allow thiscondition to be met because there is no particular point at which allthe illuminating waves have the same phase, which would be necessary inorder to meet this condition since the reference wave used is sphericaland constant. Embodiment 4, quite similar moreover to Embodiment 3,allows this condition to be met because, with an appropriate control ofthe beam deflection system, it is possible to generate illuminatingwaves whose phase at a given point is constant. Embodiment 5 also meetsthis condition.

(iii)—that the position of the optical elements is adjustedappropriately so that the phase difference between the illuminating waveand the reference wave is in fact constant. This adjustment is describedin 8.6. for Embodiment 4 and in 9.20. for Embodiment 5.

Conditions (ii) and (iii) imply that the phase difference between theilluminating wave and the reference wave is constant. It is possible todefine a virtual wave present in the object and such that its image,through the optical device modifying the wave coming from the objectbetween the object and the receiving surface, is the reference wave. Thephase difference between the illuminating wave and the reference wavemeans here the phase difference between the illuminating wave and thecomponent of this virtual wave on the wave vector of the illuminatingwave.

3.5. Phase Correction Method

When the microscope is not built so that the phase shift affecting eachfrequency sub-representation is constant, the differences between thephase shifts affecting each sub-representation must be determined andcompensated if necessary.

3.5.1. General Phase Correction Method

Part of the three-dimensional representation of the object isconsidered:

consisting of a subset A of the three-dimensional representation.

characterized by a function a(f) defined on A and conventionally zerooutside of A, where f is the spatial frequency vector and a(f) the valueof the representation on this spatial frequency.

affected by a Gaussian noise, the standard deviation of the noise on agiven frequency f being σ_(a)(f).

This sub-representation will be designated by the expression“sub-representation RA”. It will be said that A is the support of RA.

A second part of the three-dimensional representation of the object isconsidered, consisting of a subset B of the three-dimensionalrepresentation, characterized by a function b(f) defined on B, affectedby a Gaussian noise σ_(b)(f), designated by the expression“sub-representation RB.”

These two parts of the representation are phase shifted in relation toeach other and are assumed to have a non-zero intersection. From thesetwo parts will be generated a sub-representation RC defined on a setC=A∪B (C is the union of A and B) and defined by a function c(f)affected by a Gaussian noise σ_(c)(f).

It is possible to proceed in two steps:

a complex ratio between the two representations can be obtained, forexample, with the formula:$r = \frac{\sum\limits_{f \in E}\quad \frac{a\quad (f)\quad \overset{\_}{b\quad (f)}}{{\sigma_{a}^{2}\quad (f)} + {\sigma_{b}^{2}\quad (f)}}}{\sum\limits_{f \in E}\frac{{{b\quad (f)}}^{2}}{{\sigma_{a}^{2}\quad (f)} + {\sigma_{b}^{2}\quad (f)}}}$

 where the sums are over a set E of frequency vectors included in theintersection of the two sets A and B, or E⊂(A∩B) , the set E being ifnecessary limited to points for which the signal-to-noise ratio issufficiently high. The phase difference between the two representationsis the argument of r. The phase difference thus calculated is a goodapproximation of the most probable phase difference knowing the valuesof the representations RA and RB over the set E.

the representation RB may be “phase-corrected” in relation to RA bymultiplying it by the ratio r:

b(f )←r.b(f) where the sign←means that b(f) is replaced by r.b(f).

the function c can be obtained, for example, by the formula:${c\quad (f)} = \frac{\frac{a(f)}{\sigma_{a}^{2}\quad (f)} + \frac{b(f)}{\sigma_{b}^{2}\quad (f)}}{\frac{1}{\sigma_{a}^{2}\quad (f)} + \frac{1}{\sigma_{b}^{2}\quad (f)}}$

 The values thus assigned to the representation RC are the most probablevalues, knowing the phase-corrected representations RA and RB.

the function σ_(c)(f) can be obtained, for example, by the formula:$\frac{1}{\sigma_{c}^{2}\quad (f)} = {\frac{1}{\sigma_{a}^{2}\quad (f)} + \frac{1}{\sigma_{b}^{2}\quad (f)}}$

Both of the preceding operations constitute the grouping of RA and RB.

More detailed explanations on the calculation of these functions in theform of arrays are given in Paragraph 7.17.1. The formulas indicatedabove for phase correction carry out an intensity normalizationsimultaneously, which however is not indispensable.

This method makes it possible, from the two sub-representations RA andRB whose supports A and B have a non-zero intersection, to obtain asub-representation RC corresponding to the superposition of RA and RB.

If the three-dimensional frequency representation of the object must bereconstituted from many sub-representations whose phases are not known,the above method, applied iteratively, allows the grouping of all thesesub-representations. For example, it is possible to begin with a givensub-representation, and group it with a second sub-representation. Then,it is possible to begin with the sub-representation generated by thisgrouping, and group it with a further sub-representation. By repeatingthis grouping operation until all the sub-representations have beenintegrated in a overall representation, one finally obtains thethree-dimensional frequency representation of the object. The onlyconditions to be met in order for this method to succeed are:

that no sub-representation or group of sub-representations has an emptyintersection with all the other sub-representations.

that the object does not have an excessively singular frequencyrepresentation, which would be for example zero on a set of pointsseparating in two its three-dimensional frequency representation.

These conditions are easily met for all biological objects as soon as alarge number of representations is acquired.

For example:

an embodiment may be derived from the system of [Fercher] in which thethree simultaneous illuminating waves produced by the diffractiongrating have been replaced by a single illuminating wave of variabledirection. In this case, a sub-representation is constituted by atwo-dimensional frequency representation obtained for a givenilluminating wave.

an embodiment can be derived from the microscope implemented in [Turpin3] in which, in addition to the rotation of the object, variations inthe direction of the illuminating wave have been authorized. In thiscase, a sub-representation is constituted by all the two-dimensionalfrequency representations obtained for a given illuminating wave whenthe object is rotated.

in the case of Embodiment 5, Paragraph 9.19., a sub-representation isconstituted by a two-dimensional frequency representation obtained for agiven illuminating wave. A small number of sub-representations is firstgrouped in a basic representation having a non-zero intersection withall the other sub-representations obtained under similar conditions. Allthe representations are then phase-corrected in relation to the basicrepresentation, and then an overall representation is generated.

in Embodiments 3, 4 and 5, four intermediate sub-representations aregenerated each time, as explained in 7.17.1.1. These fourrepresentations are grouped in a single representation by theapplication of this general method.

According to a variant of the invention, the microscope thus comprisesmeans for:

determining, for each sub-representation RB, a coefficientcharacterizing the phase difference between this sub-representation andanother sub-representation, part of a sub-representation or group ofsub-representations RA, this coefficient being calculated from values ofRA and RB over a set included in the intersection of the supports of RAand RB.

correct the phase of RB so as to obtain for RB the same phase referenceas for RA.

The simplest way to correct the phase of RB is to carry out amultiplication by the coefficient r as indicated above. However, thiscorrection can also be carried out by physical means, in which case thephase does not need to be corrected during the calculation stage. Anexample of such an embodiment is described in 7.18.7.

The phase difference affecting each sub-representation can berecalculated with each acquisition. This is necessary in Embodiments 1and 2, for which these differences are not reproducible. In the case ofEmbodiments 3, 4 and 5, this phase difference is reproducible and canconsequently be measured during a phase preparatory to acquisition. Anexample of such an embodiment is described in 7.18.1.

According to a variant of the invention, the microscope comprises meansfor:

determining, for a given sub-representation RB, the frequencyrepresentation RC resulting from the grouping of RB with anothersub-representation RA.

determining a coefficient characteristic of the noise affecting RCdefined over the entire support of RC, obtained from a coefficientcharacteristic of the noise affecting RA and defined on the support ofRA, and from a coefficient characteristic of the noise affecting RB anddefined on the support of RB.

The methods used can differ from the formalism set forth above. Forexample, in Embodiment 1, the quantity$\frac{1}{\sigma_{c}^{2}\quad (f)}$

is assimilated with the number N of frequency representations reaching agiven point.

The calculations can be grouped: after phase correction of eachsub-representation, they can be grouped in an overall representation,without calculating each intermediate sub-representation. This is whatis done in all the embodiments to group two-dimensionalsub-representations into complete or partial three-dimensionalsub-representations.

Representations of the object can be calculated without passing formallythrough its three-dimensional frequency representation. For example, in7.17.3.3., a confocal representation of the object is generated using,for the final frequency representation, a value at each point which isthe sum of the values obtained for each representation reaching thispoint. The representation thus obtained is not strictly speaking afrequency representation of the object, but nevertheless carriesinformation on this object. It is also possible to generate realrepresentations of the refractive index or absorptivity. Theserepresentations can be generated simply through the frequencyrepresentation of the object, but it is also possible to modify thealgorithms so as not to formally use this procedure.

3.5.2. Absolute Phase Correction

The method set forth in 3.5.1. makes it possible to obtain athree-dimensional representation of the object. However, the overallphase of this three-dimensional representation remains arbitrary.

In the three-dimensional spatial representation of the object, obtainedfrom the three-dimensional frequency representation by taking theinverse Fourier transform, the complex number associated with each pointcharacterizes the absorptivity and the refractive index of theconsidered point. If the overall phase of the three-dimensionalrepresentation is chosen appropriately, the real part of the saidcomplex number characterizes the local absorptivity of the object, andthe imaginary party of said complex number characterizes the localrefractive index of the object. The overall phase is chosenappropriately when the origin of the three-dimensional frequencyrepresentation has a real value.

According to a variant of the invention, and in the case in which theorigin of the three-dimensional frequency representation is one of thepoints that have been acquired, the microscope has means for dividing,by its value at the origin, the three-dimensional frequencyrepresentation obtained by the method pointed out in 3.5.1. This makesit possible to obtain a spatial representation in which the real partand the imaginary part of the complex numbers represent respectively thelocal absorptivity and the local refractive index. This also allows theentire representation to be normalized.

When the origin of the three-dimensional frequency representation is notpart of the points that have been acquired, this operation isimpossible. The operator who views an image sees, for example, the realpart of the complex number, and must intuitively choose the total phaseof the representation so as to obtain the most contrasted possibleimage.

3.5.3. Phase Correction in Relation to the illuminating Wave

The general phase correction algorithms defined above exhibit thedrawback of being relatively complex to implement. A simplified versioncan be obtained when the acquisition system allows the acquisition ofthe non-diffracted part of the illuminating wave. This corresponds tothe origin of the three-dimensional frequency representation of theobject. This point is common to all the two-dimensional frequencyrepresentations.

The phase correction described in 3.5.1. can then be carried out inrelation to the sub-representation part constituted by this singlepoint. This correction can be grouped with the absolute correctiondescribed in 3.5.2. The total of the two corrections then is equivalentto dividing the entire sub-representation of the object obtained from agiven illuminating wave by its value at the origin. When a planefrequency image is generated as an intermediate calculation step, thisis equivalent to dividing the entire plane frequency image by its valueat the point corresponding to the non-diffracted part of theilluminating wave. According to a variant of the invention, the phasecorrection of the two-dimensional frequency representations is carriedout by dividing each sub-representation of the object by its value atthe origin of the three-dimensional frequency representation of theobject. This method is used, for example, in Embodiments 1, 2, 3 and 4.

3.5.4. Phase Correction in Relation to Pre-Recorded Phase Values

A three-dimensional frequency representation of the object is obtainedfrom a series of two-dimensional frequency representations eachcorresponding to a different illuminating beam.

Each of these two-dimensional frequency representations can be phasecorrected in relation to the sub-representation constituted by theorigin alone, as indicated in 3.5.4. However, the high intensity of thecorresponding point on each two-dimensional frequency representationmakes difficult the simultaneous acquisition of the rest of therepresentation. According to a variant of the invention, the acquisitionof plane frequency images takes place in two phases:

a preliminary phase during which are recorded the values obtained at theimage point of the illuminating wave, for each illuminating wave;

an acquisition phase proper during which the direct beam can beobstructed and during which the values of the plane frequency images arerecorded.

A two-dimensional frequency representation can then be obtained for eachilluminating wave from these two recordings, the value at the imagepoint of the illuminating wave being obtained from the first recordingand the value at any other point being obtained from the secondrecording. The method described in 3.5.4. can then be applied to eachtwo-dimensional frequency representation obtained.

When a series of two-dimensional frequency representations is obtained,for example to “film” the movement of cells, the preliminary phase mustnot be repeated. It only needs to be carried out once before the startof the acquisitions.

In order for this method to be functional, the phase difference betweenthe illuminating beam and the reference beam, at the level of thereceiving surface, must be reproducible. The illuminating beamgeneration systems used in Embodiments 3, 4 and 5 meet this condition.This phase correction phase is described for example in 7.18.1. and9.18.2.

3.6. Vibration Compensation

The method described in 3.5.4. presupposes the reproducibility of theilluminating beams. The method described in 3.5.3., in the case in whichseveral objectives are used, presupposes a constant phase shift betweenthe waves received on each of these objectives. However, in Embodiments3, 4 and 5, the vibrations can make these methods ineffective or lessrobust. In order for the results to be reliable, these vibrations mustbe compensated.

For this purpose, the system can periodically acquire a reference image.The reference image consists, for example, of an image obtained on thereceiving surface for a fixed illuminating wave, which is not modifiedwhen the illuminating wave used to obtain the “useful” plane frequencyimages varies. Each acquisition then corresponds to a reference imageacquired at a near instant. A reference image acquired at an initialinstant is chosen as absolute reference. By “useful” image is meant animage obtained on the receiving surface and for which will be calculateda two-dimensional frequency representation used to generate therepresentation of the object.

With the vector v going through the entire support of an image, wedenote by m(v) a “useful” image obtained on this receiving surface, andh(v) the corresponding reference image. We denote as h_(o)(v) thereference image chosen as absolute reference, obtained on the receivingsurface. v represents the plane frequency projection and is thus atwo-dimensional vector varying over the entire receiving surface. Byσ(v) is denoted the standard deviation of the Gaussian noise affectingthe function h(v) at each point.

The phase variation of vibratory origin can be characterized, forexample, by the coefficient$r = \frac{\sum\limits_{f}\quad \frac{h_{0}\quad (v)\quad \overset{\_}{h(v)}}{\sigma^{2}\quad (v)}}{\sum\limits_{f}\frac{{{h(v)}}^{2}}{\sigma^{2}\quad (v)}}$

which represents the most probable phase difference between thereference images h(v) and h_(o)(v).

The image m(v) can then be phase corrected as follows:

m(v)←r.m(v) where the sign←designates the sense.

When this preliminary correction has been carried out the images thusphase corrected can be used in the algorithms defined in 3.5.3. and3.5.4.

If the system is totally free from vibrations, this preliminarycorrection is not required.

If the vibrations are of the low-frequency type, the reference image canbe acquired only at a low frequency, however higher than the frequencyof the vibrations of the system.

If the vibrations are of the high-frequency type, it is possible toacquire a reference image at each useful image.

In the presence of vibrations,

this preliminary correction is essential for the application of thealgorithm defined in 3.5.4.

this preliminary correction is not essential for the application of thealgorithm defined in 3.5.3. in the case in which a single objective isused. However, if several objectives are used, it allows the fixing ofthe phase difference between the plane frequency images generated fromeach objective. In this case, it is thus also essential.

This technique is used in Embodiments 3 and 4 and described in 7.17. Itis also used in Embodiment 5 when the phase correction is carried out inaccordance with paragraph 9.18.

A variant of the invention thus consists in periodically acquiringreference images corresponding to a fixed illuminating direction, andusing these images to compensate for phase differences of vibratoryorigin affecting the plane frequency images.

3.7. Characterizing the Wave Vector of the Illuminating Wave

It is necessary to control the direction of the illuminating wave, i.e.its wave vector f_(e), for instance by mechanical means as described forexample in [Turpin]. However, these mechanical means must be veryaccurate and are costly to implement.

In the conditions defined in 3.5.3., i.e. if the acquisition systemallows the acquisition of the non-diffracted part of the illuminatingwave, the non-diffracted part of the illuminating wave corresponds tothe maximum modulus value on the frequency representation of the wavecoming from the object. According to a variant of the invention, themicroscope has means for determining the coordinates of this maximum andfor calculating, from these coordinates, the wave vector f_(e) of theilluminating wave. This method is used, for example, in Embodiments 1and 2.

However, the presence of the object can slightly distort the value ofthe wave vector thus obtained. According to a variant of the invention,the wave vectors f_(e) of each illuminating wave are obtained in apreliminary phase in which the object is eliminated or replaced by atransparent plate. The wave vectors thus obtained are thus not distortedby the presence of the object. This method presupposes that the wavevectors are reproducible from one acquisition to the other. On the otherhand, it obviates the need to calculate these wave vectors according tomechanical parameters. This method is used in Embodiments 3, 4 and 5.

3.8. Characteristics of the Receiver

3.8.1. Use of a Microscope Objective

According to a variant of the invention, the receiver comprises alarge-aperture microscope objective that transforms the rays coming fromthe object under a large aperture into paraxial rays that can bedirected towards a receiving surface. This configuration offers betterperformance than the configurations defined in [Fercher] (absence ofobjective) or [Turpin] (small-aperture objective). While [Vishnyakov]uses such a configuration without benefiting from it, owing to the useof poorly suited tomographic methods, the algorithms defined in thepresent invention make it possible to take full advantage of thisconfiguration.

3.8.2. Use of a Receiving Surface in a Frequency Plane

It is advantageous to directly acquire the image in the frequencydomain, as in [Turpin 3]. This can be done, for example, by means of thereceiver described in [Lauer 1] that allows improved performancecompared to the receiver of [Turpin 3].

3.8.3. Use of a Receiving Surface in a Spatial Plane

Besides the microscope objective itself, the receiving system defined in3.8.2. presents a paraxial part allowing the modification of the opticalsignal captured by the objective, to obtain a frequency representation.The signal from the objective first passes through the plane in whichthe objective normally forms the image of the observed sample. Thisplane will be called the spatial plane. It is then transformed by aparaxial system so that, in the plane in which the receiving surface isplaced, a plane wave coming from the object has a point image. Thisplane, in which the receiving surface is placed, will be called thefrequency plane. The paraxial part of the optical system used caninclude intermediate spatial or frequency planes. The receiving surfacecan be placed in a spatial plane, in a frequency plane, or in anintermediate plane. However, to simplify calculations, it will always beplaced either in a spatial plane or in a frequency plane. In order forthe received image to be correct, the following conditions must moreoverbe complied with:

if the receiving surface is in a frequency plane, the reference wavemust be centered virtually at a central point of the observed object.

if the receiving surface is in a spatial plane, the reference wave mustbe the image of a virtual wave which is plane on the crossing of theobserved object.

Under these conditions, the signal detected on a receiving surfaceplaced in a frequency plane is the optical Fourier transform of thesignal that would be detected in a spatial plane. One variant of theinvention constituting an alternative to the receiver defined in 3.8.2.is thus to use a receiving surface positioned in a spatial plane and areference wave which is the image of a virtual wave which is plane onthe crossing of the observed object. A numerical Fourier transform thenreplaces the optical Fourier transform.

Embodiments 1, 3 and 4 use sensor means in a frequency plane andEmbodiments 2 and 5 use sensor means a spatial plane. A plane frequencyimage can thus be obtained either directly on a receiving surface placedin a frequency plane, or by the Fourier transform of an image receivedon a receiving surface placed in a spatial plane.

3.9. Beam Attenuation

If the sensor is placed in a spatial plane, the direct illuminating wavehas, as a representation on the sensor, a constant modulus value whichis superimposed on the wave diffracted by the object. An excessivelyweak diffracted wave in relation to this constant basic level cannot bedetected correctly. On the other hand, this basic level is not very highbecause the intensity of the reference beam is spread over the entiresensor, thus generally allowing good images to be obtained. If thesensor is placed in a frequency plane, the illuminating wave isconcentrated at a point and, as in the previous case, an excessivelyweak diffracted wave in relation to this constant basic level cannot becorrectly detected. As the wave is concentrated at a point, this basiclevel is high and this limitation is troublesome.

According to an advantageous variant of the invention, a devicecontrolling the attenuation of the beam, having one or more attenuationlevels, is introduced to solve this problem. The attenuation devicemakes it possible to obtain successively several recordings differing inthe intensity of the illuminating wave. A less noisy value of thediffracted wave is then obtained by combining these recordings. Thefinal value of the diffracted wave is calculated for example at eachpoint from the recording for which the intensity of the wave received atthe considered point is highest, but for which the sensor remainsunsaturated at the considered point and in its immediate surroundingsfor all the interference patterns allowing said recording to beobtained.

Given the use of the beam attenuation device, the variant of theinvention in which a plane wave has a point image allows the detectionof weaker diffracted waves. In fact, the waves are not superimposed onthe sensor with any other wave and very low levels can be detected whenthe illuminating beam intensity is high.

Such a device is used in Embodiments 1, 3 and 4.

3.10.

Illuminating Beam Generation System

Whatever the embodiment, an illuminating beam generation method must bedesigned. The methods proposed by [Turpin] have the drawback ofrequiring significant mechanical movements and hence of greatly slowingdown the system.

An optical system, for example the one described in 8.1.1., cantransform a parallel beam having a given spatial extension and variabledirection into a parallel beam whose spatial extension has been reducedand whose directional variations have been amplified. In general, smalldirectional variations applied to a beam having a broad spatialextension can be amplified by an optical system through a reduction inthe spatial extension of the beam. As the spatial extension of theilluminating beam required for a microscope is small, this principle canbe used for the optical amplification of small mechanical movements.

A beam with a variable direction can be transformed by a lens into abeam with a variable position in the back (image) focal plane of thislens. A directional variation of the beam in a part of its optical pathis thus equivalent to a position variation in another part of itsoptical path and vice-versa. In intermediate planes, the variation is ajoint position and directional variation. There is thus no reason todifferentiate between a system generating illuminating wave positionvariations and a system generating directional variations, as thesesystems are equivalent.

According to one variant of the invention, the illuminating beamgeneration system comprises:

a beam deflector generating variations in a paraxial beam.

a large-aperture optical element (for example, a microscope objective ora condenser) transforming said incoming paraxial beam variations intosignificant directional variations in the outgoing beam.

Said system can also comprise a lens system designed so that the beam isparallel at the exit of said large-aperture optical element.

The beam deflector may, for example, be a mirror mounted on a positionerallowing the control of its orientation. This solution is implemented inEmbodiments 1 and 2. However, this solution has two drawbacks:

the movement of the mirror generates vibrations which disturb thesystem. After each movement of the mirror, it is necessary to wait forthe absorption of vibrations before proceeding with the acquisition.

the phase difference between the reference beam and the illuminatingbeam is not reproducible, thus preventing the use of certain algorithmssuch as those defined in 3.5.4.

Each of the beam deflectors described in 3.10.1., 3.10.2 and 3.10.3.enables these two problems to be solved.

3.10.1. Beam Deflector Based on a Series of Binary Deflectors

A system capable of sending the beam back in two directions can be builtby means of a birefringent prism which transmits the ordinary beam andthe extraordinary beam in two different directions. The laser beam usedmust then be polarized. A polarization rotator placed before the prismallows the orientation of its polarization in the ordinary direction orthe extraordinary direction, which implies a different angle ofdeflection by the prism. However, available ferroelectric polarizationrotators, which have the advantage of being fast, do not allow arotation of 90 degrees but a rotation of about 80 degrees. This preventsthe possibility of having both a beam polarized exactly in the ordinarydirection, for one of the positions of the polarization rotator, and abeam polarized exactly in the extraordinary direction for the otherposition. Hence, in one of the positions, is created a spurious beamdeflected in an undesired direction. To eliminate this spurious beam, itis necessary to use at the output of the birefringent prism a polarizerselecting only the desired beam. In order for this polarizer not toeliminate the beam in the other position of the rotator, a secondrotator must be introduced between this polarizer and the prism, saidsecond rotator being used to bring the electric field vector of the beamback to the passing direction of the polarizer when it is not theredirectly at the exit of the prism.

A system capable of returning a beam in many directions can beconstituted by combining in series several of these elementary systems.By combining two of them, which produce a deflection of the sameamplitude but in two different orthogonal directions, a doublet isformed. By combining N doublets in series, each doublet beingcharacterized by birefringent prisms having characteristics such thatthe deflection angle of doublet number i is proportional to 2^(i), oneobtains 2^(N) possible deflection values in each direction. For example,with N=8, there is a total of 256×256 beam deflection directions.

According to a variant of the invention, the beam deflection system isconstituted by the association of elementary deflectors in series, eachof these elementary deflectors comprising a birefringent prismdeflecting differently the ordinary beam and the extraordinary beam,preceded by an electronically controlled polarization rotator andallowing the orientation of the electric field vector of the beam alongthe ordinary axis or the extraordinary axis of said prism, and followedby a second rotator and by a polarizer allowing the elimination ofspurious beams.

Such a device is used in Embodiment 3.

3.10.2. Beam deflector based on spatial modulators

A spatial modulator is a two-dimensional matrix of pixels allowing thephase or intensity modulation of a wave in a plane. Most spatialmodulators are based on liquid crystals. Common LCD screens constitutean example of a spatial intensity modulator.

A spatial plane, on the path of the illuminating beam, will be definedby a plane in which this beam is parallel and is centered on the opticalaxis. A frequency plane will be defined as a plane in which this beamhas a point image.

A parallel beam reaching a spatial plane has, in this plane, a complexrepresentation of the form exp{j2π(f_(x)x+f_(y)y)} in which (x,y) arecoordinates of a point of the plane and in which (f_(x),f_(y)) are thecoordinates of the projection of the wave vector in this plane. If aphase modulation device is placed in this plane and if a phase shift ofthe form θ=2π(g_(x)x+g_(y)y+c) is applied by means of this device, thewave has, after passing through said device, a complex representationexp{j2π((f_(x)+g_(x))x+(f_(y)+g_(y))y+c)}. The spatial modulation devicehas thus modified the direction of the incident wave. Wave vectors thatcan generate such a spatial modulation device are included in a cone theaperture of which depends on the maximum values of g_(x) and g_(y)permitted by the modulator. This cone will be called “deflection cone.”

If an intensity modulation device is used instead of the phasemodulation device, it is possible to apply an attenuation function ofthe type cos{2π(g_(x)x+g_(y)y+c)}. After passing through the device, thewave then has a form of the typeexp{j2π((f_(x)+g_(x))x+(f_(y)+g_(y))y+c)}+exp{j2π((f_(x)−g_(x))x+(f_(y)−g_(y))y−c)}which corresponds to the superposition of two plane waves whose wavevectors are symmetrical in relation to the axis oriented along the wavevector of the wave leaving the device in the absence of modulation. Oneof the two waves can be stopped by a diaphragm, in which case the deviceconstitutes a beam deflector comparable to the preceding one.

Intermediate modulation devices performing joint phase and intensitymodulation may also be used.

A variant of the invention thus consists in using, as beam deflector, asuitably controlled spatial modulator.

According to a variant of the invention, said modulator is a phasemodulator controlled so as to generate a phase shift in a form as nearas possible to θ=2π(g_(x)x+g_(y)y).

Prior-art modulation devices operate pixel by pixel. This discretizationleads to the generation of spurious frequencies outside of thedeflection cone. One variant of the invention consists in eliminatingthese spurious frequencies by means of a diaphragm placed in a frequencyplane on the path of the wave coming from the phase modulator.

Modulation devices allowing fast modulation are binary, i.e. to a givenpixel there correspond only two possible phase or intensity values. Theuse of a binary modulation device results in the presence of a spuriousplane wave symmetrical with the wave to be obtained in relation to anaxis constituted by the direction of an undeflected beam. In the case ofbinary modulators, this is true even in the case of a phase modulator,whereas in the case of modulators generating a continuous modulation,this problem can be avoided by using a phase modulator. According to avariant of the invention, the diaphragm filtering the spuriousfrequencies is dimensioned so as to filter not only the frequencieslocated outside of the deflection cone, but also part of the frequencieslocated within the deflection cone, so as to stop the spurious planewave.

Binary modulation devices also have the drawback of generating spuriousfrequencies included within the deflection cone and constituting afrequency “noise.” According to a variant of the invention, thesefrequencies are stopped by an intensity modulator placed in a frequencyplane along the path of the beam coming from the phase modulator, andcontrolled so as to allow only the sought frequency to pass.

Such a device is used in Embodiment 4.

3.10.3. Beam deflector consisting of a mobile mirror designed so thatvibrations are not troublesome

The beam deflectors described in 3.10.1. and 3.10.2. are based on theuse of liquid crystal devices and polarizers. These devices are notavailable in the ultraviolet radiation range. To use ultraviolet rays,other means are required.

In prior-art devices, the entire system was placed on an optical table.

According to one variant of the invention, the beam deflection device ismade up of a mirror placed outside the optical table, the separationbetween the illuminating beam and the reference beam being provided by aseparator fixed on the optical table and positioned after said mirror onthe path of the beam.

As the mirror is placed outside the optical table, it does not generatevibrations in this table. As the separation of the beams takes placeafter the mirror, its vibrations also do not generate phase shiftsbetween illuminating beam and reference beam. This consequently solvesthe problem of vibrations.

On the other hand, the fact that beam separation takes place after themobile mirror means that the reference beam varies at the same time asthe illuminating beam. This variation must be taken into account in thedesign of the system and compensated. For example, if the receivingsurface is placed in a spatial plane, variations in the direction of thereference wave result in translations of the plane frequency image.According to a variant of the invention, this effect is compensated byproviding a translation in the opposite direction of the plane frequencyimages obtained.

This technique is used, for example, in Embodiment 5.

3.11. Compensation of errors due to polarization

The image generation method described by [Wolf] is based on a scalardiffraction theory and assumes that the wave passing through the objectis diffracted isotropically in all directions by each point of theobject.

It is only on the basis of this assumption that the theoreticalresolution of λ/4 can be obtained. Scalar diffraction theory is howevernot valid for high diffraction angles. The intensity diffracted by apoint of the object depends on the direction of the diffracted wave, thedirection of the illuminating wave, the polarization direction of theilluminating wave and the polarization direction of the diffracted wave.

3.11.1. Compensation by real-coefficient multiplication

The wave diffracted by the object differs from the wave which would bediffracted if the diffraction were isotropic by a real multiplicativefactor depending on the:

illuminating wave propagation direction

illuminating wave polarization

diffracted wave propagation direction

diffracted wave polarization

According to one variant of the invention, the microscope comprisesmeans for determining this multiplicative factor and compensating it bymultiplying the received waves by the inverse of said factor. Such avariant of the invention is described in 7.18.8.

3.11.2. Compensation in the case of an anisotropic material

If the observed object is made up essentially of an anisotropicmaterial, the effects of the diffraction differ from what they are in anisotropic material. The material is then characterized at each point bysix crystalline parameters plus absorptivity.

In the particular case in which the observed object is a uniaxialcrystal, the refractive index of the ordinary ray has a constant value.According to one variant of the invention, a three-dimensionalrepresentation in which the complex numbers obtained characterize theabsorptivity and the ordinary index of refraction can be calculated.According to a variant of the invention, this representation iscalculated from plane frequency images obtained for an illuminating wavepolarized so that it constitutes an ordinary beam.

As the ordinary polarization direction varies with the wave'spropagation direction, it is necessary to be able to modify thepolarization direction of the illuminating wave. However, as thephenomena are linear, one need only record the wave received at everypoint for two polarization directions of the illuminating wave to beable to deduce the wave diffracted by the object for an illuminatingwave of any direction. According to a variant of the invention, themicroscope comprises means for generating two polarization directions ofthe illuminating wave. According to a further variant of the invention,the microscope also comprises means for analyzing the diffracted wavealong two polarization directions, thus making it possible todistinguish the ordinary diffracted wave from the extraordinarydiffracted wave. According to this variant of the invention, the planefrequency images are then obtained from four elementary imagescorresponding to each combination of two polarization directions and twoanalysis directions.

Such a variant of the invention is described in 7.18.9. In thedescription of 7.18.9. only the analysis direction corresponding to theordinary ray is used, but it is also possible to take into account theanalysis direction corresponding to the extraordinary ray. Two planefrequency images are then obtained for each illuminating wavepropagation direction, one corresponding to the ordinary index and theother to the extraordinary index. The final frequency representation isobtained from this set of images, taking into account the variations inthe extraordinary index at each point of the plane frequency imagescorresponding to the extraordinary index.

3.11.3. Compensation by combining several polarization and analysisdirections

In the case of the isotropic material, the method described in 3.11.1.has the drawback of causing the noise level to rise considerably. Oneway to avoid this problem is to acquire at least four plane frequencyimages corresponding to each combination of two distinct polarizationsof the illuminating wave and two distinct polarization directions of thediffracted wave. An appropriate algorithm then makes it possible, fromthese four images, to calculate a single image corresponding to thescalar parameter sought. According to a variant of the invention, themicroscope consequently comprises means for generating two distinctilluminating wave polarizations, and means for analyzing the diffractedwave along two distinct polarization directions. A variant of theinvention is that the microscope comprises means for calculating, fromthe plane frequency images corresponding to each combination of twopolarization directions and two analysis directions, a single planefrequency image representing a complex scalar quantity complying withthe condition of uniform diffraction in all directions. This principleis used in Embodiments 3, 4 and 5. The principle for calculating saidscalar quantity is described in detail in 7.12.1.

In the case of wavelengths in the visible, said means for varying theilluminating wave polarization can consist of a liquid crystalpolarization rotator. Said means for varying the diffracted waveanalysis direction can consist of a polarization rotator associated witha polarizer. In the case of ultraviolet, these devices are notavailable.

3.11.4. Variation of illuminating wave polarization direction in the UVrange

In the UV range, polarization rotators can be replaced by a quartzretardation plate rotating around an axis by mechanical means. However,these mechanical movements slow the system down considerably. For thisreason, use is made of a system in which the only mechanical movementsare those of shutters and in which the beam to be closed off has aspatial extension which is as small as possible so that the movement ofthe shutter is as small as possible. It is then possible to use ahigh-speed shutter or a shutter system with a rotating crownwheel.

According to one variant of the invention, such a system includes:

a beam separator which breaks the beam down into a beam A and a beam B.

lenses placed on each beam A and B and focussing these beams on pointsof focus at which the shutters are placed.

a device making it possible to again superimpose beams A and B havingpassed through their respective shutters.

a device placed on the place of one of the beams A or B, in the part ofthe path in which the two beams are distinct, and modifying thepolarization of this beam.

Such a system can also comprise additional lenses designed to reformparallel beams after passage through the shutters. It can also comprisea second polarization modification device. The beam polarizationmodification device can be a retardation plate. The beam separation andbeam superposition devices can be semi-transparent mirrors. Such adevice is used in Embodiment 5.

3.11.5. Variation of analysis direction in the UV range

The wave coming from the object can be broken down into a wave whoseelectric field vector is parallel to that of the reference wave and awave whose electric field vector is orthogonal to that of the referencewave. The intensity received on the receiving surface is the sum of theintensity of the wave whose electric field is orthogonal to thereference wave and the intensity produced by the interference of thereference wave and the wave whose electric field is parallel to thereference wave. The first of these intensities does not depend on thephase of the reference wave and consequently does not modify the complexvalue of the wave coming from the object measured by the combination ofinterference patterns corresponding to different phases of the referencewave. It is thus only the wave whose electric field vector is parallelto that of the reference wave which is obtained on the receivingsurface.

The analysis direction of a wave can consequently be modified simply bymodifying the polarization direction of the reference wave or,symmetrically, by modifying the polarization direction of the wavecoming from the object.

According to a variant of the invention, the analysis direction ismodified by varying the polarization direction of the reference wave orof the wave coming from the object.

According to a variant of the invention, the polarization of thereference wave or of the wave coming from the object is modified by adevice comprising:

a beam separator breaking the beam down into two beams A and B

a retardation plate LA placed on the path of beam A and a retardationplate LB placed on the path of beam B, the angle between the neutralaxes of these two retardation plates being 45 degrees.

The polarization direction of the beam having passed through retardationplate LA is then deduced from the polarization direction of the beamhaving passed through retardation plate LB by a rotation whose angle isthe angle between the neutral axes of retardation plates LA and LB. Ifthis angle is 45 degrees, the beams coming from plates LA and LB willalways have orthogonal polarizations whatever the polarization directionof the incident beam.

The two beams A and B can then be combined by a superposition deviceafter having passed through shutters, as in the case of the devicedescribed in 3.11.4. This has the drawback of requiring the use ofshutters at a point of the beam path in which the direction of the beamis variable and in which the beam can consequently not be focussed on afixed point.

According to a variant of the invention, the two beams A and B areseparately superimposed on the beam coming from the object (if theyconstitute the reference wave) or the reference wave (if they come fromthe object). The interference patterns corresponding to eachpolarization direction are then formed on two distinct receivingsurfaces.

Whether the plane frequency images corresponding to each polarizationare obtained on distinct receiving surfaces or on the same receivingsurface, a phase difference is then created between these images andmust be compensated. According to a variant of the invention, theretardation plates are positioned so that the reference and illuminatingbeams reaching the receiving surface have different polarizationdirections, preferably 45 degrees from each other. Then, the part of thewave coming from the object which comprises frequencies near theilluminating wave is detected on the two receiving surfaces and can beused to calculate said phase difference.

3.12. System for eliminating direct illumination

The illuminating wave is generally much more intense than the diffractedwave. It can saturate the sensors or reduce considerably thesignal-to-noise ratio of the system by requiring acquisition of highsignal levels. The illumination of the non-diffracted part of theilluminating wave during the acquisition phase or during part of theacquisition phase clearly improves the performance of the system. In afrequency plane, the non-diffracted part of the illuminating wave has apoint image and can be eliminated by placing an absorbing element onthis point. According to one variant of the invention, the systemaccordingly comprises a device for eliminating the non-diffracted partof the illuminating wave, placed in a frequency plane, and absorbing thebeam on a small zone around the point corresponding to this illuminatingwave.

According to a variant of the invention, this device is made up of aspatial intensity modulator controlled to allow passage at all pointsexcept on a limited zone around the impact point of the non-diffractedpart of the illuminating wave. This variant is implemented in Embodiment4.

According to a further variant of the invention, this device comprises aglass which is mobile in translation within a frequency plane, anabsorbing black spot placed on the glass being designed to stop thedirect beam, the position of the glass being controlled so that thisblack spot coincides with the point of impact of the non-diffracted partof the illuminating wave. This variant of the invention is implementedin Embodiment 5.

3.13. Spatial modulator utilization device

The spatial modulators used in 3.10.2. or in 3.12. may in particular behigh-speed binary modulators operating by reflection. They are generallyused by means of a polarizing semi-transparent mirror which is a deviceof cubic form sending back incident beams in two different directionsdepending on their polarization. This device, owing to its thickness,generates a slight aberration which widens the point corresponding to agiven frequency, which is detrimental to the quality of the imagesobtained.

To avoid the use of this device, it is possible to use beams coming fromthe modulator under an oblique angle. The incident and reflected beamsare then separated. However, this method deforms the distribution of thegenerated frequencies. To prevent this deformation, said oblique anglemust be small.

According to one variant of the invention, this problem is solved byusing a device consisting of a mirror with two orthogonal reflectingfaces and a lens traversed in one direction by the beam directed towardsthe modulator and in the other direction by the beam reflected by themodulator. The incident beam is reflected by one side of the mirror,passes through the lens, is reflected by the modulator, passes againthrough the lens in the opposite direction and is reflected by thesecond side of the mirror, resuming its initial direction. The incidentand reflected beams can be partially superimposed on the lens, but areseparated on the two-sided mirror. In order to obtain this separation onthe mirror, while maintaining as small an oblique angle as possible, thetwo-sided mirror must be positioned approximately in one focal plane ofthe lens and the modulator must be positioned in the other focal planeof the lens.

3.14. Use of objectives traversed in both directions and/or of severalobjectives

The object can be illuminated on one side and the wave coming from theobject can be picked up on the opposite side by an objective, thusallowing the reconstitution of part of the frequency representation ofthe object. However, other parts of the frequency representation of theobject can be constituted only from the wave moving towards the side ofthe object from which the illuminating waves come. According to onevariant of the invention, an objective is associated with an opticalsystem allowing, on the one hand, the measurement on a sensor of thewave coming from the sample and having gone through the objective and,on the other, the formation of an illuminating wave which, aftercrossing the objective, becomes in the sample a plane illuminating waveof variable direction. This objective is then crossed in one directionby the illuminating wave moving towards the object, and in the otherdirection by the diffracted wave coming from the object. It plays boththe role of objective receiving the wave coming from the object and therole of high-aperture system transforming the small directionalvariations of the wave coming from the beam splitter into largedirectional variations of the wave in the object. This can be achievedfor example by means of a semi-transparent mirror placed on the path ofthe beam coming from the object and superimposing the beam coming fromthe object and directed in a given direction on the illuminating beammoving in the opposite direction. This variant of the invention isimplemented in Embodiments 3, 4 and 5 which comprise several objectives.One arrangement in which only one objective is used is described in7.18.10.

Possible limitations having to do with the direction of the illuminatingwave, characterized by its frequency vector, influence the performanceof the system. Maximum precision is obtained when all the possibledirections are used. Similarly, it is desirable to record the wavediffracted by the object in all directions. When a single microscopeobjective is used, its aperture limits the directions in which it ispossible to record the wave diffracted by the object. According to anadvantageous variant of the invention, several objectives focussed onthe sample are used, which makes it possible to record the wave comingfrom the sample along more directions. The objectives then cover almostthe entire space around the sample, and the illuminating waves mustnecessarily pass through these objects to reach the sample. According toone variant of the invention, each objective is associated with anoptical system allowing, on the one hand, the measurement on a sensor ofthe wave coming from the sample and having passed through the objectiveand, on the other, the formation of an illuminating wave which, afterpassing through the objective, becomes a variable direction planeilluminating wave in the sample. The illuminating waves can thus begenerated in all the directions covered by the aperture of theobjectives, and similarly the waves coming from the object can bemeasured in all these directions. Considering a variant of theinvention, the acquisition and calculation system takes into account allthe waves measured on all the sensors for all the illuminations usedand, from these data, generates the three-dimensional frequencyrepresentation of the object. Each pair (illuminating wavedirection-direction of wave from object) corresponds to a point of thefrequency representation of the object, and the frequency representationthus generated consequently comprises all the points obtainable from theilluminating waves and the diffracted waves respectively produced andreceived by all the objectives.

A large number of objectives can be used in order to receive all thewaves coming from the sample, or in order to increase the workingdistance by using low-aperture objectives. However, most of the samplesobserved in practice are flat and can be placed conveniently between twocover glasses. By virtue of a variant of the invention constituting thebest compromise between utilization difficulty and performance, use ismade of two large-aperture microscope objectives positioned oppositeeach other, the flat sample being introduced between these twoobjectives. This solution is used in Embodiments 3, 4 and 5. InEmbodiments 1 and 2, which exhibit lower performance but are easier tofabricate, a single microscope objective is used.

3.15. Generation of inverse beams

When two or more microscope objectives are used, a plane frequency imageis generated from the wave received by each objective. Each point of aplane frequency image corresponds to a given wave vector of thediffracted wave. In order to calculate the three-dimensional frequencyrepresentation of the object, it is necessary to determine these wavevectors correctly, and this in a coordinate system common to the wavevectors received by each objective.

Knowing the K factor and the optical center, defined in [Lauer 1],allows the determination of the wave vectors corresponding to each pointof a frequency image. However, the coordinate system used fortwo-dimensional representations RA (before translation of vector −f_(e))reconstituted from the plane frequency image obtained from a givenobjective is different from that used for the representations RBobtained from the facing objective. To establish a correspondencebetween these two coordinate systems it is necessary to determine thecoordinates of certain points in the coordinate system used for RA aswell as in the coordinate system used for RB.

Each point PA of the representation RA is the image of a wave vectorf_(e) of the illuminating wave which reaches this point in the absenceof an object, and has the coordinates of this wave vector. To thisvector there corresponds a wave vector −f_(e) of opposite directionwhose image is a point PB of representation RB. The coordinates of pointPB in a coordinate system used for RA are the opposite of thecoordinates of point PA in this coordinate system.

The correspondence between the two coordinate systems can thus beestablished if the coordinates of point PB are also determined incoordinate system RB. This can be done by generating, by optical means,a wave vector beam opposite to the illuminating beam and by determiningthe coordinates of the image point of this beam in the coordinate systemused for RB. If this correspondence is established at a sufficientnumber of points, the relationship between the coordinate systems usedfor RB and RA can be easily determined and these representations can bemodified to use a common coordinate system.

In like manner, it is possible to obtain, by optical means, a directcorrespondence between coordinate systems RB and RA. This requires theadjustment of a certain number of optical elements. To perform thisadjustment, it is possible to check continuously the correspondencebetween the coordinates of point PB obtained in each of the coordinatesystems used, and this for a certain number of points PB (three pointsin principle).

In both cases, it is necessary to generate a beam having the samecharacteristics as the illuminating beam, but propagating in theopposite direction. In general, given a beam used in the system, theterm “opposite indicator beam” will designate a beam having the samecharacteristics but propagating in the opposite direction.

Considering one variant of the invention, the microscope consequentlycomprises, during the 35 adjustment phase, means for generating anindicator beam opposite to the illuminating wave. These means may beeliminated after the adjustment phase corresponding to thedetermination, by calculation means or optical means, of thecorrespondences between coordinate systems RB and RA.

According to a variant of the invention, the microscope also comprisesmeans for generating, during an adjustment phase, an indicator beamopposite to the reference beam. This beam will also be used in certainadjustment phases. For example, if the receiving surface is in afrequency plane, the reference wave is centered virtually at a centralpoint of the object. The indicator beam opposite to the reference wavemakes it possible to adjust the position of the objectives so that theseobjectives are focussed on the same point.

According to a variant of the invention, when the receiving surface is aspatial plane, an additional beam centered on this spatial plane is alsoused during an adjustment phase, as is its opposite indicator beam. Thisbeam makes it easy, for example, to adjust the position of theobjectives in the absence of a reference wave centered on a point of theobject.

In Embodiments 3, 4 and 5, each beam used has an opposite indicatorbeam, and the means of generating these opposite indicators aredescribed as forming part of the microscope and are not eliminated afterthe adjustments have been completed: shutters are simply used to cut offthese beams.

According to one variant of the invention, the device generating anopposite indicator beam from an original beam comprises:

a semi-transparent mirror separating the original beam into anunmodified beam and a secondary beam.

a lens focussing the secondary beam in a focussing plane.

a mirror placed at the point of focus, which reflects the opposite beamback to said lens.

The fact that the mirror is placed at the point of focus guarantees thatthe reflected beam has exactly the opposite direction of the incidentbeam. The reflected beam passes through the lens again in the oppositedirection. The part of this beam which is then reflected again by thesemi-transparent mirror has the same characteristics as the unmodifiedbeam but is directed in the opposite direction.

3.16. Determination of differences in the position of objectives

From waves coming from the object and passing through a given objective,it is possible to generate a three-dimensional representation of theobserved object. In spatial representation, this representation isrelative to a given origin, that will be called characteristic point ofthe objective. In general, characteristic points of the objectives useddo not coincide. Consequently the part of the frequency representationgenerated from an objective is translated in relation to that obtainedfrom another objective. This translation results in a frequencymodulation, the points in the frequency space obtained from a givenobjective consequently being affected by a variable phase shiftcorresponding to this modulation. Considering a particular variant ofthe invention, the microscope comprises means for compensating thistranslation and generating a representation of the object in which thephase shift affecting each point of the representation is constant. Tobe able to superimpose the representations obtained from each objective,and in accordance with one variant of the invention, the microscopecomprises means for determining the coordinates of the characteristicpoints of each microscope objective, in a common coordinate system. Itis then possible to translate appropriately each representation beforesuperimposing them. This translation in the spatial domain is equivalentto a demodulation in the frequency domain, which can be carried outdirectly on the plane frequency images.

3.16.1. Determination of coordinates of characteristic points of eachobjective

According to a variant of the invention, this can be obtained by usingthe beam centered on a central point of the observed object and itsopposite indicator beam. This beam is received on one sensor afterhaving passed through the objectives, and its opposite indicator isreceived on another sensor. From the beam received on a sensor, thetwo-dimensional frequency representation of this beam can be obtainedand the coordinates of its point of focus can be determined. The pointof focus of the beam is the same as that of its opposite indicator beam.The difference between the coordinates of the point of focus of the beamobtained from one objective and those of its opposite indicator obtainedfrom another objective is equal to the difference between thecoordinates of the characteristic points of these objectives in a commoncoordinate system.

This method can be implemented with any number of objectives, providedthat the configuration is such that no group of objectives is opticallyisolated, i.e. that a given objective group, if it does not include allthe objectives used, can always be reached by a beam coming from anobjective outside the group. For example, if six objectives are used,they must be grouped two by two: each objective must receive beamscoming from the other two objectives.

This aspect of the invention is implemented in 7.9.1. and in 9.12.

3.16.2. Determination of displacements of each objective

It is generally necessary to move the objectives in order to introducethe observed object. This operation modifies the coordinates obtainedand calls for the repetition of the preceding operation. However, thepresence of the object disturbs the beam which passes through it andprevents any precise result from being obtained. According to onevariant of the invention, this problem is solved by using parallel beamsof variable direction. A parallel beam has a point plane frequency imageand the value obtained at this point is affected little by the localirregularities of the observed sample.

The difference between the phase of such a beam received on a sensorbefore the movement of the objectives and the phase of the same beamafter the movement of the objectives depends on the vectorcharacterizing the displacement of the characteristic point of theobjective receiving the beam in relation to the characteristic point ofthe objective from which the beam comes. From these phase differencesestablished for a sufficient number of beams, it is possible by means ofan appropriate algorithm to determine this displacement. According tothis variant of the invention, the phase of a set of parallel beamsreaching a given sensor is consequently measured a first time in theabsence of the object and a second time in the presence of the object.From the phase differences and possibly from the intensity ratios thusmeasured, an appropriate algorithm can recalculate the displacement ofthe characteristic points of each objective. These phase and intensitydifferences can be characterized, for each parallel beam, by a complexvalue obtained as the quotient of the value obtained in the presence ofthe object by the value obtained in the absence of the object at acorresponding point of the plane frequency image. This value will becalled the phase and intensity ratio on a given parallel beam.

Knowing the initial coordinates of the origin of each representation andits displacement, its current coordinates can be deduced and thedifference in position can be compensated. This method can beimplemented with any number of objectives, with the same provision as in3.16.1.

The first measurement in the absence of the object is carried out, forexample, in 7.9.2. and in 9.13. In these two cases, the correctionrelated to the position values determined as indicated in 3.16.1. isdescribed in the same paragraph as that measurement.

The second measurement in the presence of the object and the calculationof the displacements are carried out for example in 7.11. and in 9.15.In both cases, the absolute positions are calculated directly withoutintermediate displacement determination, the correction related to theposition values having already been performed as indicated above. Thisdetermination is coupled with the calculation of the average index ofthe object whose principle is given in the next paragraph.

3.17. Determination of index and thickness of the object

3.17.1. Principle

When two microscope objectives opposite to each other are used and whenthe observed object forms a layer between two flat cover glasses, theaverage index of the object, if it differs from the nominal index of theobjective (index of optical liquid designed to be used with theobjective), creates a spherical aberration which distorts thethree-dimensional representations obtained. This spherical aberration isreflected in phase and intensity variations of the beams measured in3.16.2., these variations depending on the index and thickness of theobject. According to one variant of the invention, a program uses thevalues measured in 3.16.2. to determine simultaneously the displacementsof the objectives, and the index and thickness of the object.

The calculation carried out in 7.11. or in 9.15. is an embodiment ofthis variant of the invention.

3.17.2. Minimization algorithm

For given values of displacements and of the index and thickness of theobject it is possible to calculate the phase and intensity ratios oneach parallel beam. According to a particular variant of the invention,the calculation algorithm determines the values of the displacements andthe index and thickness of the object which minimize the standarddeviation between the theoretical values thus calculated and the valuesactually measured.

This algorithm must determine an absolute minimum on a set of fiveparameters of a noisy function, the standard deviation functionexhibiting, even in the absence of noise, local minimum values otherthan the absolute maximum. This problem consequently lends itself poorlyto conventional minimization methods.

When the values of the parameters are known approximately, the algorithmcan maximize, in a first phase, the value at the origin obtained bycompensating the phase differences due to these parameters. Maximizationof a function being equivalent to a minimization of its opposite, theterm maximization will be used only to define the algorithm, but itwould also be possible to speak of minimization.

According to a variant of the invention, the algorithm comprisesiterative phases during which it determines, for each iteration, anabsolute maximum on a set of vales of the parameters varying discretely,the function to be maximized having previously been filtered toeliminate the frequencies that would cause aliasing when sampling at thevariation intervals of the parameters. According to this variant of theinvention, the interval is reduced with each iteration and the centralpoint of the set on which the parameters vary during an iteration is themaximum determined in the preceding iteration.

Such an algorithm makes it possible in general to converge towards thesolution despite the local maxima and the large number of parameters.

Such an algorithm is described in 7.8.2.

3.18. Object position determination

The exact position of the object has no influence on the values measuredin 3.16.2. and can consequently not be obtained from these values. Onthe other hand, it can modify the three-dimensional representationsobtained: the spherical aberration affecting a given three-dimensionalrepresentation depends on the position of the object. If the index ofthe object differs from the nominal index of the objectives, thisposition must be determined.

The position of the object in relation to the objectives affects theproper superposition of the phase corrected two-dimensionalrepresentations. If it is not correctly evaluated and taken into accountas a correction factor, abnormal phase differences appear between pairsof phase corrective two-dimensional representations, on the part of thefrequency space that corresponds to the intersection of theserepresentations.

According to a variant of the invention, the measurement of the positionof the object in relation to the objectives comprises an acquisitionphase during which a series of plane frequency images are acquired,corresponding to a series of illuminating beams of variable orientation.Knowing the position parameters, the index and the thickness of theobject, previously calculated by the maximizing algorithm described in3.17., and knowing the sought position, a program can determine thetwo-dimensional representations corresponding to each plane frequencyimage. According to this variant of the invention, the program fordetermining the position of the object in relation to the objectives isconsequently a minimization program that determines the value of theposition parameter that minimizes the abnormal phase differences. Such aprogram is described in detail in paragraph 17.15.

3.19. Use of objectives exhibiting aberrations

The design of an objective without aberrations is difficult. Aberrationsincrease in proportion to the size of the optical elements used, so thatit is difficult to obtain a great working distance. It is also difficultto obtain a high numerical aperture.

In the present microscope, the complex value of the wave received on areceiving surface is recorded. The frequency representation of part ofthe wave coming from the object can be reconstituted when this part ofthe wave coming from the object reaches the receiving surface. Thefrequency representation of the wave coming from the object is obtained,in every case, by a linear relationship from the wave reaching thereceiving surface. The only “indispensable” property of the objective isconsequently its ability to pick up a significant part of the wavecoming from the object, and transform it into a paraxial beam reachingthe receiving surface. An objective having this property can easily bedesigned and have a high numerical aperture and long working distance.According to one variant of the invention, an objective affected byaberrations greater than the limit imposed by the diffraction is used,and the calculation program reverses the linear relationship between thewave coming from the object and the wave picked up on the receivingsurface so as to compensate these aberrations.

According to a variant of the invention, the microscope objective isdesigned to comply with the following property:

(1)—The aberration affecting the image formed in the image plane of theobjective must be less than a fraction of the diameter of the observedpart of this image.

The requirement (1) ensures that, for the major part of the objectstudied, the entire beam coming from a given point reaches the receivingsurface. In fact, the spatial extent of the observed sample is limited,depending on the systems, by a diaphragm or by the size of the sensorused. In the presence of spherical aberration, when the distance betweena point and the limit of the observed area is less than thecharacteristic distance of the spherical aberration, part of the wavescoming from this point do not reach the receiving surface and the imageof this point can consequently not be reconstituted with precision. Ifthe area concerned remains small in size, this drawback is not verytroublesome: this is what is guaranteed by compliance with condition(1). If the area concerned were too large, the precision would beaffected on the entire image.

Requirement (1) is similar to the usual requirement concerning sphericalaberration, but is considerably reduced. In fact, results of goodquality can be obtained with a spherical aberration of the order of 10or so wavelengths, whereas in a conventional microscope sphericalaberration must be a fraction of a wavelength.

However, an objective not having certain additional properties may bedifficult to use. In fact, the linear relation linking the wave receivedon the receiving surface to the frequency representation of the wavecoming from the object may be relatively complex. The algorithmiccompensation of aberrations may thus require high computation volumes.According to one variant of the invention, this problem is solved byusing an objective having, in addition, the following property:

(2)—The image, in the back (image) focal plane of a beam which isparallel in the observed object, must be a point image.

The plane frequency image used in all the embodiments is equivalent tothe image formed directly in the rear focal plane. The requirement (2)means that each point of a plane frequency image corresponds to a givenfrequency of the wave in the object. If requirement (2) were notcomplied with, it would be necessary to use a generalized algorithmconsisting in obtaining the value associated with a given point as alinear combination of the values detected on a set of neighboringpoints. Requirement (2) is relatively simple to comply with. The use ofan objective not complying with condition (2) would only be of limitedinterest, but would considerably complicate the calculations needed toobtain an image. An example of the use of an objective complying withconditions (1) and (2) is described in paragraph 7.21.

In the case of an objective complying only with conditions (1) and (2),the image reconstitution algorithm must take into account the relationbetween the coordinates of a point on the plane frequency image and thecorresponding frequency of the wave in the object.

By B(α) is denoted the image point, in the back focal plane, of aparallel beam in the object and forming in the object an angle a withthe optical axis.

According to a variant of the invention, the algorithms are simplifiedby using an objective complying in addition with the followingcondition:

(3)—The distance between point B(α) and point B(0) must be proportionalto sin α.

Compliance with condition (3) means that the coordinates of a point onthe plane frequency image must be directly proportional to thecomponents along two corresponding axes of the wave vector of the wavein the object, which considerably simplifies the calculations. If anobjective complies with conditions (1)(2)(3), spherical aberrationresults only in a phase shift of each point of a given frequencyrepresentation, a phase shift which can then be easily compensated. Anexample of the use of an objective complying with conditions (1)(2)(3)is described in 7.20. Condition (3) is roughly equivalent to thecondition of Abbe's sines.

In an objective complying only with conditions (1)(2)(3), the image isdisturbed in the vicinity of the diaphragm over a distance equivalent tothe characteristic distance of the spherical aberration. Thisdisturbance is eliminated by the use of a conventional objective rid ofany spherical aberration, which thus makes it possible in principle toobtain the best images. However, the use of an objective complying onlywith (1)(2)(3) allows a significant reduction in the design requirementsof the objective. This makes it possible to obtain an objective with alonger working distance, a less expensive objective, or an objectivewith a higher numerical aperture. This technical solution may thus bepreferable in certain cases.

The use of such a microscope objective requires the algorithmiccompensation of phase differences affecting the plane frequency imageand which are the consequence of spherical aberration. This calls forthe determination of the function characterizing the sphericalaberration induced by the microscope. Because the objective exhibits asymmetry of revolution, the phase difference affecting a point of aplane frequency image depends only on the distance between this pointand the optical center. The spherical aberration may thus becharacterized by a one-dimensional function representing the phasedifference as a function of this distance.

According to a variant of the invention, this function can be obtainedby the optical calculation program used for the design of the objective.In fact, this program allows the tracing of beams and can easily beimproved to also enable optical path calculations. As the phasedifferences are proportional to the optical path differences affectingthe rays coming from a given point, they can be deduced from geometricalconsiderations of this type.

According to a further variant of the invention, this function can bemeasured by optical means. In accordance with this variant of theinvention:

two identical microscope objectives facing each other are used.

use is made of an illuminating wave centered on a point of the plane atwhich a first objective normally forms the image of the observed sample.

the wave received in the plane at which the second objective forms theobserved sample's image is detected.

In the absence of spherical aberration, the phase of the detected wavemust be constant. In the presence of spherical aberration, the phasedifference due to the aberration is twice the difference due to a singlemicroscope objective. This makes it possible to obtain the soughtfunction.

3.20. Compensation for spherical aberration and position differences

Spherical aberration due to the refractive index of the object,spherical aberration due to the properties of the objective, andpositioning errors relative to the objectives, all result in phasedifferences applied to the plane frequency image. These phasedifferences must be corrected in order to obtain a good quality image.

According to a particular variant of the invention, this corrected iscarried out by multiplying each plane frequency image by a correctionfunction taking into account the various parameters determined above.The calculation of such a function is described in 7.16. and themultiplication operation is carried out for example in step 2 of thealgorithm described in 7.17. If an objective exhibiting sphericalaberration is used, the calculation described in 7.16. must be modifiedas indicated in 7.20.

3.21. Regular sampling

When two microscope objectives are used, the sampling interval on theplane frequency image generated from one of the receiving surfaces canbe taken as the basis for the sampling interval of the three-dimensionalrepresentation of the object along two corresponding axes. If noprecaution is taken:

The image points of the illuminating waves on this plane frequency imagedo not correspond to integer values of the coordinates in pixels.

If two objectives are used, the sampling interval and the axes on theplane frequency image generated from a receiving surface associated withthe objective located opposite do not correspond to the samplinginterval and to the axes of the three-dimensional representation of theobject.

The result of this is that the sampling of the three-dimensionalrepresentation of the object is not regular. According to one variant ofthe invention, this sampling is made regular along two axescorresponding to the axes of the plane frequency representations. Thequality of the three-dimensional representation of the object is thenclearly improved.

The plane frequency image can be modified in particular, due toimperfections in the optical system, by rotation or by similitudetransformation. To obtain a regular sampling, it is necessary to cancelor compensate for these imperfections.

Further, in Embodiment 4, there must be a point to point correspondencebetween the different SLMs used and the CCDs. To obtain thesecorrespondences, adjustments of the same type are needed.

According to a variant of the invention, the microscope consequentlycomprises one or more optical devices allowing a rotation adjustment ofthe images generated in the frequency planes and/or one or more devicesallowing adjustment by magnifying the images generated in the frequencyplanes.

3.21.1. Adjustment of representation scale (homothetic transformation)

This adjustment is in fact a magnification adjustment. According to avariant of the invention, the magnification of an image is adjusted bymeans of an optical system with a variable focal length. Such a systemmay, for example, be composed of two lenses, a variation in the distancebetween said lens resulting in a variation in the overall focal length.Such a device is used in Embodiments 4 and 5 and is described in8.1.4.1.

3.21.2. Image rotation adjustment

According to a variant of the invention, said adjustment is carried outby means of a device consisting of a first group of fixed mirrors and asecond group of mirrors, complying with the following conditions:

the first group of mirrors symmetrizes the wave vector of the incidentbeam in relation to a given axis.

the second group of mirrors symmetrizes the wave vector of the incidentbeam in relation to a second axis.

the second group of mirrors is mobile in rotation around an axisorthogonal to the plane of these two axes.

Both groups of mirrors then have the effect of imparting a rotation tothe beam represented in a frequency plane, the angle of rotation beingtwice the angle between the two axes of symmetry. Such a device is usedin Embodiments 4 and 5 and is described in 8.1.4.2.

3.22. Phase shift system

The phase shift system used can be a piezoelectric mirror, constitutingthe most usual solution. However, used at high speed, such a mirrorgenerates vibrations. According to an advantageous variant of theinvention, the phase shift system used is a birefringent blade inducinga phase shift of 120 degrees between its neutral axes, preceded by apolarization rotator allowing the orientation of the electric fieldvector of the beam along one or the other of said neutral axes, andfollowed by a second polarization rotator for bringing the polarizationdirection of the beam at the output of the device back to its directionat the input of the device. As this system allows only a phase shift of120 degrees, it is necessary to combine two in series to obtain a phaseshift of −120, 0, or +120 degrees.

3.23. Data processine method in the case of a limited RAM

Three-dimensional frequency representation calculations involve largequantities of data. As these data are normally accessed in a randomorder, they cannot be stored during calculations on a sequential-accessmedium such as a hard disk and must be stored on a random-access mediumsuch as an internal computer memory (RAM).

According to an advantageous variant of the invention, adapted to thecase in which the system does not have sufficient RAM to allow storageof all the data, the calculation algorithm is modified so as to processthe data block by block, a block corresponding to a large amount of datawhich can then be stored sequentially on a sequential-access medium andloaded into central memory only during the processing time of saidblock. For this purpose:

The modified algorithm carries out, within a three-dimensional space,processing operations horizontal plane by horizontal plane, eachhorizontal plane being stored on the sequential-access medium in asingle block.

In order to be able to also carry out processing along the verticaldimension, the algorithm incorporates axis exchange phases which make itpossible to bring the vertical axis temporarily into a horizontal plane.

The axis exchange procedure works block by block, the blocks generallyhaving equal or similar dimensions along the two axes to be exchangedand having as byte size the maximum size that can be stored in thecentral memory of the system (random-access memory—RAM).

This method is implemented in Embodiment 1 and described in paragraph5.21.

3.24. Images venerated by the microscope

The three-dimensional representations generated by the presentmicroscope can be stored and transmitted in the form of athree-dimensional array of complex numbers. According to one variant ofthe invention, it is possible to generate two-dimensional projections orsections representing either the refractive index or absorptivity in theobject.

In the case of a projection, one generates a projection of thethree-dimensional image on a projection plane and along a projectiondirection orthogonal to the projection plane. Each of the projectionplane is obtained from all the values of the three-dimensional spatialrepresentation which is located on a line passing through this point anddirected along the projection direction.

According to a variant of the invention, the value associated with eachpoint of the projection plane is obtained by extracting the maximumvalue of the real or imaginary part or the module from the points of thethree-dimensional spatial representation located on the correspondingline.

According to a variant of the invention, the value associated with eachpoint of the projection plane is obtained by integrating the complexvalue of the points of the three-dimensional spatial representationlocated on the corresponding line. It is then possible to display eitherthe real part or the imaginary part of the projection thus obtained.According to this variant of the invention, the projection can beobtained more rapidly as follows, in two steps:

step 1: extraction, in frequency representation, of a plane passingthrough the origin and orthogonal to the projection direction.

step 2: inverse Fourier transformation of this plane.

The two-dimensional array thus obtained constitutes a projection alongthe direction having served to extract the frequency plane.

3.25. Optical element positioning system

The embodiments described require the use of many high-precisionpositioners. These positioners are costly elements ill suited to massproduction and capable of losing their adjustment with time.

According to a variant of the invention, this problem is solved by usingremovable positioners during the manufacture of the microscope, eachelement being positioned and then fixed by a binder, for example anadhesive, and the positioner being removed after final solidification ofthe binder.

3.26. Shocks, vibration and dust protection system

The microscopes described consist of a set of elements fixed to anoptical table. During possible transport, shocks, even minor ones, canlead to the misadjustment of the system. During prolonged use, dust maydeposit on the different optical elements.

According to one variant of the invention, the greater part of theoptical device is included in a hermetically closed box which is itselfincluded in a larger box, the link between the two boxes being providedby shock absorbers placed on each side of said hermetically closed box.This system protects the microscope from shocks and from dust whileproviding a good suspension for the optical table.

4. BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1 to 24 relate to a first embodiment. FIG. 1 is a general diagramof the microscope optical system. FIGS. 2 and 3 represent in detail theangular positioner (110) already represented in FIG. 1. FIG. 4 is ageneral diagram of the vertical mechanical and anti-vibration support ofthe microscope. FIG. 5 represents in detail a tensioner of FIG. 4. FIG.6 represents a dimensioning example of the optical part. FIGS. 7 to 9and 15 and 16 are graphic representations serving as a support for theexplanation of the microscope's operating principle. FIG. 10 representsthe algorithm of a program allowing the adjustment of piezoelectriccontrol voltages (122). FIG. 11 represents the detailed algorithm of animaging procedure used in the preceding program. FIG. 12 represents thealgorithm of a program for adjusting the intensity attenuator consistingof the polarizer (105) and of the polarization rotator (104) and forobtaining its characteristics. FIG. 13 represents the detailed algorithmof a “low-level” imaging procedure used in the preceding program and inthe 2D or 3D image acquisition programs. FIG. 14 represents thealgorithm of a focussing program for obtaining a 2D image and focussingthe microscope objective (113). FIG. 17 is a support for theexplanations concerning the adjustment of the condenser. FIG. 18represents the algorithm of the three-dimensional image acquisitionprogram. FIG. 19 represents the algorithm of a calculation programgenerating, from the results of the acquisition, a three-dimensionalrepresentation of the object. FIG. 20 represents schematically anoperation carried out by the first part of this program. FIG. 21represents in detail the algorithm of this first part. FIG. 22represents schematically an operation carried out by the second part ofthis program. FIG. 23 represents the algorithm of a third part of thisprogram. FIG. 24 represents the algorithm of a last part of thisprogram.

FIGS. 25 and 26 relate to a second embodiment. FIG. 25 is a generaldiagram of the optical part of the microscope. FIG. 26 represents thealgorithm of the “low-level” image acquisition procedure used.

FIG. 71 illustrates a specific device used in the adjustment operationsfor Embodiments 3 to 5.

FIGS. 27 to 59 relate to a third embodiment. FIGS. 27 and 28 constitutea general diagram of the optical part of the microscope. FIG. 29 showsschematically the optical path of the rays between the objective and thesensor. FIG. 30 represents the beam attenuation device used. FIG. 31 isa schematic diagram illustrating its operation in the case in whichthere is actually an attenuation. FIG. 32 is a schematic diagramillustrating its operation in the case in which there is no attenuation.FIG. 33 represents the phase shift device used. FIG. 34 is a schematicdiagram illustrating its operation. FIG. 35 continues FIG. 34 indicatingthe phase differences of the various vectors. FIG. 36 continues FIG. 34indicating the modulus of the different vectors. FIG. 37 represents abasic unit of the beam deflection device used. FIG. 38 represents thebeam deflection and switching device, formed by the association of thesebasic units. FIGS. 39 and 40 are schematic diagrams explaining theoperation of a basic unit. FIG. 39 corresponds to one deflectiondirection and FIG. 40 to the other deflection direction possible. FIG.41 illustrates the calculation of the deflection of the beam by a prism.FIGS. 42 to 44 illustrate steps in a phase rotator marking procedure.FIG. 42 illustrates the first step and FIGS. 43 and 44 illustrate asecond step, in two different cases. FIG. 45 illustrates the calculationof the path difference produced on a parallel beam by an object with agiven refractive index and thickness. FIG. 46 illustrates thecalculation of the path difference produced on a parallel beam by thedisplacement of the origin of the reference wave in relation to whichthe optical path is calculated. FIGS. 47 to 50 and FIG. 60 illustrate analgorithm for calculating the refractive index and thickness values ofthe sample as well as the displacement of the origin of the referencewave. FIG. 47 corresponds to the highest level of this algorithm andFIG. 50 to the lowest level. FIG. 51 represents, in a three-dimensionalspace, different vectors used to evaluate the effect on the diffractedwave of the polarization of the illuminating beam. FIG. 52 represents,in the plane of a sensor, vectors deduced from the preceding. FIG. 53represents an algorithm for obtaining beam deflector control indicesbased on the coordinates of a point of direct impact sought for theilluminating beam. FIG. 54 illustrates the calculation of the pathdifference of a wave coming from a point of the object in relation to areference wave. FIG. 55 illustrates the calculation of the pathdifference between a wave coming from a point of the object and thereference waves used on the two sensors of the system. FIG. 56illustrates the path, on one of the sensors, of the point of directimpact of the illuminating wave during an imaging procedure. FIG. 57represents an algorithm determining the position of the object inrelation to the objectives. FIG. 58 illustrates how thethree-dimensional frequency representation of the object is obtained bythe superposition of two-dimensional representations. FIG. 59 representsa section of the three-dimensional frequency representation of theobject.

FIGS. 61 to 70 and 72 relate to a fourth embodiment of the invention,constituting the preferred embodiment. FIGS. 61, 62 and 63 constitute ageneral diagram of the optical part of the microscope. FIGS. 64 and 65illustrate the operation of the beam deflector used in this embodiment.FIG. 66 illustrates the operation of a direct illuminating waveeliminating system. FIGS. 67 and 68 illustrate the principle used tocontrol the beam paths. FIGS. 69 and 70 illustrate a system used toobtain controlled rotation of a beam. FIG. 72 illustrates the image tobe obtained in one of the adjustment operations.

FIGS. 73 to 82 relate to a fifth embodiment. FIGS. 73 and 74 constitutea general diagram of the optical part of the system. FIGS. 75 and 76illustrate images obtained during an intermediate calculation step. FIG.77 illustrates an algorithm used to determine the control sequence of amirror deflecting the beam and to determine the control sequence of aglass stopping the direct illuminating wave. FIGS. 78 and 79 representimages obtained during an adjustment step. FIG. 80 illustrates theprinciple used to vary the polarization of the illuminating wave and theanalysis direction. FIG. 81 represents the points corresponding todifferent illuminating waves used in a phase correction algorithm. FIG.82 represents a diaphragm used in this embodiment.

FIGS. 83 to 85 relate to a device for positioning and fixing in adefinitive manner the optical elements. FIG. 83 and FIG. 84 representparts of this device, and FIG. 85 represents the entire device.

FIGS. 86 to 88 relate to a device designed to protect the microscopefrom shocks and dust. FIG. 86 represents a schematic diagram of thedevice. FIGS. 87 and 88 represent a specific configuration suited to thecase of Embodiment 4.

FIGS. 89, 90 and 91 relate to the use of non-stigmatic objectives. FIG.89 represents an embodiment of such an objective. FIG. 90 is used as asupport for the statement of conditions to be complied with by thisobjective. FIG. 91 shows a plane frequency image obtained by means ofthis objective.

5. DESCRIPTION OF A FIRST EMBODIMENT

This embodiment is the simplest and is not costly.

5.1. Material characteristics

A laser beam with a 633-nm wavelength polarized in the directionorthogonal to the figure is emitted by the helium-neon laser (100) andpasses through the filter (101) made up of zero, one or more stackedfilters in tinted glass of the Schoff type. It is then separated into anilluminating beam Fe and a reference beam Fr by the semi-transparentmirror (102). The illuminating beam then passes through a filter (103)of the same type as (101), then through a polarization rotator (104)based on ferroelectric liquid crystals and marketed by the companyDisplaytech Inc., 2602 Clover Basin Dr., Longmont, Colo. 80503, UnitedStates. It then passes through a polarizer (105). The beam then goesthrough an achromat (106) then a diaphragm (107) and an achromat (108).It is reflected by a mirror (109) fixed on the angular positioner (110)controlled by two stepping motors. It then passes through an achromat(124) and then a condenser (111) and reaches the object (112). Thecondenser is, for example, a Nikon aplanatic/achromatic condenser of 1.4aperture designed for a light source located at infinity. The object(112) is a sample placed between plate and cover glass, the absorptivityand refractive index variations of which are relatively small, and athree-dimensional image of which is to be obtained. The achromat (124)is placed as near as possible to the lowest lens of the condenser (111),possibly in the condenser body. The front focal point of the achromat(124) coincide with the rotation center of the mirror (109). Lenses(106) and (108) must be such that the laser beam focuses in the frontfocal plane of the assembly made up of the condenser (111) and theachromat (124), and have a sufficient aperture upon its arrival in thisfocal plane. The beam is thus again parallel when it reaches the object(112). The object plane must be horizontal so that the optical oilrequired to use the objective and the immersion condenser does not flow.The optical axis is thus vertical and the elements of FIG. 1 are fixedon a vertical plate (300) shown in FIG. 4.

The angular positioner (110) is shown in detail in FIGS. 2 and 3. FIG. 2represents a plate on which the mirror (109) is glued, in a bottom view.FIG. 3 represents the entire device seen from the side. The plate (200)is placed on mobile contact pads (205) and (212) whose contact pointswith the plate are (203) and (204), and a fixed contact pad whosecontact point is at (201), the plate (200) being slightly hollowed atthis point. It is maintained by a spring (210) fixed to the plate at(202) and to a fixed pad (211). The mobile contact pad (205) movesvertically in translation and is incorporated in a single-axis actuatorof conventional type: it is blocked in rotation by a flexible element(207) and is driven in translation by the rotation of the rod (206)coming out of the stepping motor (208) which is, for example, a 400steps/rev. stepping motor. The drive is obtained, for example, by meansof a screw thread. The contact pad (212) is likewise driven by therotation of the motor (213). The fixed contact point (201) must be inthe optical axis of the objective (113) and the center of the mirrormust be on this contact point. The motors (208) and (213) are fixed on aplate (209) itself fixed on the support plate (300). The center ofrotation of the mirror, which must be positioned on the optical axis, isa point of the reflecting surface of the mirror located on a lineorthogonal to the plane of the mirror and passing through the center of(201).

The wave coming from the object (112) passes through the microscopeobjective(113). This objective is an aplanatic objective (giving a flatimage of a plane), with a high aperture (for example 1.25), of theimmersion type, and forming a magnified image of the object at a finitedistance.

In the plane in which the objective normally forms the image of theobject to be observed, a diaphragm (114) is inserted and allows spatialfiltering of the image. Behind this plane is positioned an achromat(115) whose front focal plane must coincide with the rear focal plane ofthe objective (113). A second achromat (117) whose rear focal plane isin the plane of a CCD sensor (118) forms in the plane of this CCD theimage of the rear focal plane of the objective (113). The CCD (118) isincorporated in a camera (119) delivering an analog video signal and apixel clock.

The reference beam first passes through a filter (120) of the same typeas (101) and is then reflected by a mirror (121) mounted on the mobileend of a piezoelectric translator (122). It then passes through a lens(123) which focuses the beam at a point. The diverging beam coming fromthis point is reflected partially by the semi-reflecting mirror (116),thus superimposing it on the beam coming from the object, enabling therecording of their interferences on the CCD (118). The point of focus ofthe beam coming from the lens (123) must have a virtual image afterreflection on the semi-transparent mirror (116) at the center of theimage of the diaphragm (114) through the achromat (115). Thepiezoelectric translator (122) permits the modulation of the referencebeam phase.

The manual positioners used in the system are not represented in thefigure. The microscope objective (113) is mounted on a focussing device.The laser (100) is mounted on a two-axis positioner permitting thedirection to be adjusted. The fixed part of the piezoelectric stack(122) is mounted on a two-axis positioner allowing rotation in relationto an axis orthogonal to the plane of the figure and passing through thecenter of the mirror (121), and in relation to a second axis located inthe plane of the mirror (121), orthogonal to the first axis and passingthrough the center of the mirror (121). The condenser is mounted on athree-axis translation positioner. The camera (119) is mounted on athree-axis translation positioner. The angular position of thesemi-transparent mirrors (102) and (116) can be adjusted manually. Thelens (106) can be translated along its axis. The object (112) is fixedon a two-dimension positioner enabling it to be moved in a horizontalplane. The diaphragm (114) can be moved in a horizontal plane.

The assembly is fixed on the support plate (300), on the side of theplate opposite the viewpoint of FIG. 4. This plate is fixed to twotriangular plates (301) and (302) themselves fixed to a square base(303). The plate (300) is also fixed directly to the square base (303).Plates (300)(301)(302)(303) are in rigid aluminum alloy AU4G, forexample 20 mm thick. The attachment of the plates is possible by meansof screws and tapped holes, and must be done in a sufficient number ofpoints to ensure perfect overall rigidity. This makes it possible tokeep the system vertical while providing sufficient rigidity. Theassembly is placed on an anti-vibration support consisting of a graniteplate (304) 30 mm thick placed on a van inner tube inflated to a lowpressure (305) which dampens the vibrations and which is itself placedon a rigid wooden table (306). A rigid wooden frame (311) is fixed ontop by means of uprights (307) (308)(309)(310) to the table (306). Theentire wooden construction is reinforced so as to be perfectly rigid.The top of the plate (300) is connected by tensioners(312)(313)(314)(315) to the corners of the rigid frame. Each tensioner,shown in detail in FIG. 5, consists of a set of rubber bands (316)tensioned between two rings (318)(317), these rings themselves beingfixed to the plate (300) and to the frame (311) by means of cords(320)(319), the assembly being placed under tension. The inner tube(305) makes it possible to have, for the suspended AU4G assembly, smallresonance frequencies for translation movements, of the order of 2 Hz.The tensioners (312) to (315) allow the swing of the assembly to belimited. The swinging frequency can be evaluated simply by imparting alight swinging movement to the assembly and measuring the time requiredto have, for example, ten swings. The swinging frequency is adjusted bymodifying the number of rubber bands used for each tensioner. The higherthe number, the higher the swinging frequency. It must be adjusted sothat the swinging frequency is of the same order as the resonancefrequency for translation movements, i.e. about 2 Hz.

The mounting of the different elements on the plate (300), and inparticular the mirrors and the semi-transparent mirrors, must be carriedout so as to ensure maximum overall rigidity. All usual precautions mustbe taken so as to limit vibrations.

The following description relates to a particular practical designexample for the device. The distances mentioned are represented in FIG.6. The dimensions given are approximate and some of them must becorrected in an adjustment phase. The lens (106) is an achromat with afocal length of 10 mm. The lens (108) is an achromat of 24-mm diameterand 120-mm focal length. The distance D7 between (106) and (108) is 250mm. The distance D6 between the lens (108) and the center of the mirror(109) is about 100 mm. The distance D5 between the center of rotation ofthe mirror (106) and the achromat (124) is 120 mm. The achromat (124) isplaced about 5 mm from the lowest lens of the condenser, in the body ofthe condenser. The condenser is a clear-bottom achromatic/aplanaticimmersion type condenser with an aperture of 1.4, for example, the Nikonmodel. The microscope objective is an ×100 planachromatic objective withan aperture of 1.25, finite distance, forming the image 160 mm from theshoulder of the objective, with a focal length of about 1.8 mm, forexample the Zeiss model. The distance D4 between the shoulder of theobjective and the diaphragm (114) is 160 mm. The distance D3 between thediaphragm (114) and the achromat (115) is 20 mm. The achromat (115) hasa focal length of 200 mm and a diameter of 30 mm and its most curvedside is oriented towards the semi-transparent mirror (116). The achromat(117) has the same characteristics and its most curved side is alsooriented towards the mirror (116). The distance D2 between the twoachromats is 85 mm, allowing the insertion of a semi-transparent mirror(116) of sufficient dimensions. The distance between the achromat (117)and the CCD (118) is 200 mm. The lens (123) has a diameter of 4 mm and afocal length of 6 mm. The distance D9 between this lens and the opticalaxis is about 70 mm. The distance D8 between the achromat (115) and thecenter of the semi-transparent mirror (116), located on the opticalaxis, is about 45 mm. The laser (100) is a helium-neon laser with awavelength in vacuum λ=633 nm, polarized in the directional orthogonalto the figure, with a power of about 0.5 mW, and a beam diameter of 0.5mm. The CCD sensor is a square pixel sensor, the surface of the pixelbeing about 8.5×8.5 micrometers, and the useful surface in pixels havingdimensions at least equal to 512×512 pixels. The camera delivers a CCIRvideo signal and a pixel clock, and its exposure time is equal to halfthe duration of a field, i.e. {fraction (1/50)} second for anon-interleaved CCIR camera whose fields last {fraction (1/25)} second.This gives a delay between the end of a field and the start of the nextexposure period, a delay which can be used to modify the illuminationconditions without any influence by the transition on the image. Thepiezoelectric positioner (122) is a prestressed piezoelectric ‘stack’ inthe form of a cylinder whose body is fixed and whose end moves 15micrometers for an applied voltage of 100 volts.

The calculation system is, for example, a ‘PC’ type computer equippedwith appropriate acquisition and control cards and possibly withadditional computation resources, operating for example under theWindows 95 operating system. The video signal acquisition card,operating in real time, samples the signal on 8 bits and acquires imagesof hpix×vpix size in which hpix and vpix are higher than 512 andmultiples of 4. The pixels are sampled according to the pixel clock,hence corresponding exactly to the pixels of the CCD. The piezoelectricpositioner is controlled directly by a digital/analog conversion carddelivering a signal included, for example, between zero and U_(max), forexample with U_(max)=10 volts. A resistance is inserted between theoutput of the conversion card and the terminals of the piezoelectricactuator so as to limit the current. Its value is set so that the risetime of the voltage at the terminals of the actuator (122) is about 1ms. The polarization rotator (104) is equivalent to a half-wave platewhose axis can turn and has two equilibrium positions separated by anangle of 45 degrees. If it is positioned so that the polarized beam isparallel to this axis in one of the equilibrium positions, it will causethe beam to turn 90 degrees in the other position. The polarizationrotator is controlled by applying a bipolar voltage, −5 V, correspondingto one equilibrium position and +5 V to the other. Each terminal of thepolarization rotator is connected to an 0/5 V output of a digital outputcard, and its two positions are controlled by applying in one case 0 Vto one output and 5 V to the other, and reversing for the otherposition. The stepping motors (208) and (212) are also controlled bycomputer, via an appropriate control card and electronics. The computeris equipped with sufficient internal memory (at least 32 MB) and a harddisk of sufficient size (at least 4 GB).

5.2. Miscellaneous conventions

The following conventions will be used in the rest of this description,as well as in the other embodiments:

The letter j represents sometimes an index, sometimes the pure imaginarycomplex number of modulus 1. If there should be any ambiguity, thecomplex number j will be denoted {tilde over (j)}.

The sign=symbolizes the assignment of a value right of the sign to asymbol left of the sign, or equality, depending on the situation.

The expression a+=b means a=a+b

a multiplied by b is written ab, a.b or a*b

The modulus of a complex number z is denoted |z| and its conjugate isdenoted {overscore (z)}.

The expression a%b will mean <<a modulo b >>

If a is a boolean integer, {overscore (a)} is its complement, hence{overscore (0)}=1 and {overscore (1)}=0

The discrete Fourier transform, in its most usual form, transforms aDirac located at the origin into a constant and transforms a constantinto a point located at the origin. The discrete Fourier transform usedthroughout this patent transforms a constant into a Dirac located in themiddle of the transform array and transforms a Dirac located at thispoint into a constant. This means that the ‘zero’ in frequency or inposition is placed at the middle of the array and not at the origin ofthe array. This modified transform is obtained from the usual form bycarrying out, before and after the transformation, a permutation ofindices. An array E of dimension fdim is transformed as follows:

1-first permutation: E[i]=E[(i+fdim/2)%fdim]

2-usual Fourier transformation of Array E

3-inverse permutation: E[i]=E[(i+fdim/2)%fdim]

where the sign % designates the modulo.

The two-dimensional Fourier transform of an array of lines and columnsis obtained by performing the one-dimensional Fourier transformationdefined above on each line of the array, thus generating an intermediatearray, and then performing this transformation on each column of theintermediate array to obtain the transformed array.

Similarly, the three-dimensional Fourier transform consists in carryingout successively, on each axis, one-dimensional transforms extended tothe entire array.

5.3. Opera tiding principles

The value of the light wave coming from the object under a givenillumination at a point of the CCD sensor is obtained from the recordingof three interference patterns received on the sensor, the phase of thereference wave being shifted 120 degrees between each of these patterns.

If s is the light vibration coming from the object and r is the lightvibration constituting the reference wave during the first recording,the total light vibrations reaching the sensor during the threesuccessive recordings are:${s_{0} = {s + r}},{s_{1} = {s + {re}^{j\frac{2\quad \pi}{3}}}},{s_{2} = {s + {{re}^{{- j}\frac{2\quad \pi}{3}}.}}}$

The intensities recorded successively are thus:

 |s ₀|² =|s| ² +|r| ²+(s{overscore (r)}+{overscore (s)}r)${s_{1}}^{2} = {{s}^{2} + {r}^{2} + \left( {{s\overset{\_}{r}^{{- j}\quad \frac{2\quad \pi}{3}}} + {s\overset{\_}{r}^{{- j}\quad \frac{2\quad \pi}{3}}}} \right)}$${s_{2}}^{2} = {{s}^{2} + {r}^{2} + \left( {{s\overset{\_}{r}^{{- j}\quad \frac{2\quad \pi}{3}}} + {s\overset{\_}{r}^{{- j}\quad \frac{2\quad \pi}{3}}}} \right)}$

It is possible to reverse these formulas, thus obtaining:${s\frac{\overset{\_}{r}}{\quad {r}}} = {{\frac{1}{6\quad {r}}\left( {{2{s_{0}}^{2}} - {s_{1}}^{2} - {s_{2}}^{2}} \right)} + {j\quad \frac{1}{2\sqrt{3\quad}{r}}\quad \left( {{s_{1}}^{2} - {s_{2}}^{2}} \right)}}$

The above value is the light vibration coming from the object alone, thephase reference being conventionally equal to zero for a vibration inphase with the reference wave. This calculation consequently allows thereconstitution of the complex value of the wave from the intensityrecordings.

By construction, each point of the CCD sensor corresponds to a purefrequency f_(c) of the wave coming from the sample. By optical center ofthe sensor is meant the point of the sensor illuminated by a rayentering the objective along a direction strictly parallel to its axisof symmetry, and by C_(x),C_(y) is denoted its coordinates in pixels. Asthe objective complies with the sine condition, the deflection at theexit of the objective of a ray originating at a central point of theobject is proportional to the sine of its angle of entry into theobjective, which is equal to${{\sin \quad \phi} = \frac{\sqrt{f_{x}^{2} + f_{y}^{2}}}{\sqrt{f_{x}^{2} + f_{y}^{2} + f_{z}^{2}}}},$

where (f_(x),f_(y),f_(z)) are the coordinates of the spatial frequencyvector of the beam at the entry of the objective, whose norm is 1/λwhere λ is the wavelength in the observed medium, and whose direction isthat of the beam. As the rest of the optical system is paraxial, thedeflection of the point of arrival of the beam on the sensor in relationto the optical center of the sensor is also proportional to this value,and hence the coordinates (i−C_(x),j−C_(y)) of this point in relation tothe optical center are proportional to (f_(x),f_(y)). If the sensor hassquare pixels, the frequency vector at the entry of the objective of theray which illuminates the point of the sensor having coordinates (i,j)is thus equal to:${f_{c}\left( {i,j} \right)} = {\frac{1}{K\quad \lambda}\left( {{i - C_{x}},{j - C_{y}},\sqrt{K^{2} - \left( {i - C_{x}} \right)^{2} - \left( {j - C_{y}} \right)^{2}}} \right)}$

where K is a constant to be determined. This frequency vector will becalled “characteristic frequency” of the corresponding point.

Knowing the characteristic frequency of each point of the sensor and thelight vibration coming from the object and received at this point, onethus obtains, for a given illumination of the sample, the frequencyrepresentation of the wave coming from this sample.

When a sufficiently thin sample which is not highly absorbing and whichexhibits low index variations is traversed by a parallel laser beam,each point of the sample is subjected to a light vibration Ae^(j2πf)^(_(e)) ^(.r) where f_(e) is the spatial frequency vector of theilluminating beam and r is the position vector at the considered pointof the object, the origin being taken at the virtual origin of thereference wave. The absorptivity and index variations of the sampleresult in the appearance of a secondary wave which is superimposed onthe illuminating wave. A small volume dV creates a secondary waveAe^(j2πf) ^(_(e)) ^(.r)u(r)dV where u(r) is a complex coefficient whosereal part depends on the local absorptivity of the sample and whoseimaginary part depends on the index of the sample. The three-dimensionalimage of the sample generated by this microscope is the set of thevalues u(r) at each point of the sample.

When a point of the sample, of position vector r, emits a lightvibration s(r) locally, the light vibration received on the sensor at apoint of characteristic frequency f_(c) is then equal to s(r)e^(−j2πf)^(_(c)) ^(.r). A small volume dV of the object, of position vector r,illuminated by a plane wave having a spatial frequency f_(e), thuscreates, at the point of sensor having a characteristic frequency sensorf_(c), a vibration Ae^(j2π(f) ^(_(e)) ^(−f) ^(_(c)) ^().r)u(r)dV whichis superimposed on the main vibration. Integrated over the entireobject, the vibration received at a point of the sensor is hence equalto: s(P)=∫∫∫Ae^(j2π(f) ^(_(e)) ^(−f) ^(_(c)) ^().r)u(r)dV. Thisvibration is thus an element of the Fourier transform of the functionu(r), corresponding to the Fourier frequency f_(t)=f_(e)−f_(c). Theprinciple of this microscope is to record this vibration for a set offrequencies f_(e) and f_(c), and then to reconstitute the frequencyrepresentation of u(r) and finally u(r) by inverting the Fouriertransform.

The samples studied are not highly absorbent and exhibit smallvariations in the refractive index. Consequently, the illuminating waveremains very intense and illuminates a point of the CCD, which will becalled the point of impact of the illuminating wave and which is thepoint of the CCD at which the received light vibration is highest. Ifthis point has the coordinates (imax,jmax) then the frequency of theilluminating wave is:$f_{e} = {\frac{1}{K\quad \lambda}\quad \left( {{{imax} - C_{x}},{{jmax} - C_{y}},\sqrt{K^{2} - \left( {{imax} - C_{x}} \right)^{2} - \left( {{jmax} - C_{y}} \right)^{2}}} \right)}$

FIG. 7 shows the set (500) of all characteristic frequenciescorresponding to the points of the sensor. This set is a sphere portionof radius 1/λ, limited by the aperture of the objective, centered on theoptical axis (501) of the system. An example of illuminating frequencyvector (502) is superimposed on this set. Finally, the set (503) of thecorresponding frequencies f_(t)=f_(c)−f_(e) is deduced therefrom. Whenthe illuminating frequencies (502) are made to vary along a circular arc(504) as indicated in FIG. 8, each position of the illuminatingfrequency vector corresponds to a set of frequencies f_(t) forming aportion of a sphere. FIG. 9 shows a section of a set of such sphereportions (505) (506) and others, generated when the illuminatingfrequency vector is moved on the circular arc (504). When the movementof the illuminating frequency vector on the circular arc (504) becomescontinuous, a volume is generated whose sectional view is represented inFIG. 15. In a top view, the circular arc (504) is represented by thesegment (1102) of FIG. 16. When the extremity of the illuminatingfrequency vector passes through several circular arcs so as to generatea path represented in a top view in FIG. 16, the generated volume isroughly that which would be generated by rotating the surface (1100)around the vertical axis (1101). One thus obtains a three-dimensionalfrequency representation (and no longer a two-dimensional representationas in the case in which a single recording is used). From thisthree-dimensional representation in the frequency space, it is possibleto generate the function u(r) by taking the inverse Fourier transform.

To obtain the frequency representation of u(r) from the set oftwo-dimensional frequency representations, at each point will becalculated the average value of the two-dimensional frequencyrepresentations reaching this point.

The calculation of the frequency representation of u(r) and then finallyof u(r) can be carried out for example in six steps:

Step 1: For each illuminating wave, the two-dimensional frequencyrepresentation of the received wave is determined, which is a sphereportion of a three-dimensional space.

 From a point of the CCD sensor having the coordinates (i,j), oneobtains a point of the representation of the wave coming from theobject, this point having the frequency${f_{c}\left( {i,j} \right)} = {\frac{1}{K\quad \lambda}\left( {{i - C_{x}},{j - C_{y}},\sqrt{K^{2} - \left( {i - C_{x}} \right)^{2} - \left( {j - C_{y}} \right)^{2}}} \right)}$

 and having as complex value the complex value obtained at theconsidered point of the CCD sensor by the formula${s\frac{\overset{\_}{r}}{\quad {r}}} = {{\frac{1}{6\quad {r}}\left( {{2{s_{0}}^{2}} - {s_{1}}^{2} - {s_{2}}^{2}} \right)} + {j\quad \frac{1}{2\sqrt{3\quad {r}}}\quad {\left( {{s_{1}}^{2} - {s_{2}}^{2}} \right).}}}$

Step 2: For each illuminating wave, one determines the frequency f_(e)of the illuminating wave by determining the coordinates (imax,jmax) ofthe point of maximum intensity on the CCD sensor and applying theformula:$f_{e} = {\frac{1}{K\quad \lambda}\quad \left( {{{imax} - C_{x}},{{jmax} - C_{y}},\sqrt{K^{2} - \left( {{imax} - C_{x}} \right)^{2} - \left( {{jmax} - C_{y}} \right)^{2}}} \right)}$

Step 3: For each illuminating wave, one translates the representationobtained after Step 1, of a vector−f_(e) where f_(e) is the frequency ofthe illuminating wave obtained after Step 2.

Step 4: For each illuminating wave, the two-dimensional frequencyrepresentation obtained after Step 3 is divided by its value at thepoint of coordinates (0,0). This step constitutes the phase correctionoperation described in 3.5.3. and is indispensable so that thetwo-dimensional frequency representations can be superimposed in acoherent manner.

Step 5: All the two-dimensional frequency representations obtained afterStep 4 are superimposed, obtaining the frequency representation of u(r).The value assigned to a point not reached is 0 and the value assigned toa point reached is the average of the values at this point of eachtwo-dimensional frequency representation reaching this point.

Step 6: A three-dimensional inverse Fourier transformation of thefrequency representation is carried out, finally obtaining the functionu(r) in spatial representation.

In practice, the steps will be carried out in a different order and in amodified form in order to optimize the calculation time and to limit therequired memory space. The method actually used includes two steps,equivalent to the preceding six steps:

Acquisition phase: For each illuminating wave, this phase determines theplane frequency image obtained on the CCD sensor by applying, at eachpoint, the formula${s\frac{\overset{\_}{r}}{\quad {r}}} = {{\frac{1}{6\quad {r}}\left( {{2{s_{0}}^{2}} - {s_{1}}^{2} - {s_{2}}^{2}} \right)} + {j\quad \frac{1}{2\sqrt{3\quad {r}}}\quad {\left( {{s_{1}}^{2} - {s_{2}}^{2}} \right).}}}$

 Also determined are the coordinates (imax,jmax) of the point of maximumintensity on the CCD sensor, and the entire frequency plane imageobtained on the CCD sensor is divided by its value at the point ofcoordinates (imax,jmax).

Three-dimensional calculation phase: From each illuminating wavecharacterized by the values (imax,jmax), and from each point ofcoordinates (i,j) of the sensor, is obtained a point of thethree-dimensional frequency representation with the frequency$f_{t} = {{\frac{1}{K\quad \lambda}\left( {{i - C_{x}},{j - C_{y}},\sqrt{K^{2} - \left( {i - C_{x}} \right)^{2} - \left( {j - C_{y}} \right)^{2}}} \right)} - {\frac{1}{K\quad \lambda}\quad \left( {{{imax} - C_{x}},{{jmax} - C_{y}},\sqrt{K^{2} - \left( {{imax} - C_{x}} \right)^{2} - \left( {{jmax} - C_{y}} \right)^{2}}} \right)}}$

 When a point is not reached, a zero value is assigned. When it isreached several times, the average of the values obtained each time isassigned to it. When this operation has been carried out for all theilluminating waves and all the points of the sensor, thethee-dimensional inverse Fourier transform can be taken.

This method raises a practical problem, namely that the pointilluminated directly by the beam passing through the sample isilluminated much more intensely than the points corresponding to thediffracted wave. The three-dimensional representation generated whenimaging contains basically the frequencies near that of the illuminatingbeam, the other frequencies being smothered in the noise. To remedy thisdrawback, use is made of a device allowing controlled attenuation of thebeam. The part of the frequency representation corresponding to thefrequencies on which the intensity is high is obtained with a strongattenuation, and that corresponding to the other frequencies is obtainedwith a weak attenuation. The values obtained with a strong attenuationare then multiplied by a complex coefficient characteristic of the phaseshift and of the ratio of the illuminating wave amplitudes between thetwo positions of the controlled attenuation device. This controlledattenuation device consists of the polarization rotator (104) and thepolarizer (105).

The three-dimensional representation obtained from images of sizehpix×vpix corresponds to large file sizes. In order to limit the size offiles and the calculation time, the size of the images will be dividedby two by averaging during the three-dimensional representationacquisition procedure. This is equivalent to grouping the pixels 4 by 4,a group of 4 pixels on the original image being equivalent to an actualpixel used for the calculation. The size of the observable object is ofcourse reduced accordingly. The values of C_(x), C_(y) and K are dividedby 2 to take into account the new coordinate system. This lniitation inthe size of the observed image is of course optional.

The use of the microscope proper comprises:

a focussing phase on the sample, described in paragraph 5.17.

a condenser position adjustment, described in paragraph 5.18.

a filter adjustment, described in paragraph 5.19.

the acquisition phase for the standard two-dimensional frequencyrepresentations, described above, and described in detail in paragraph5.20.

the three-dimensional calculation phase, described above, and describedin detail in paragraph 5.21.

a display phase described in paragraph 5.22.

Before being able to use this microscope, several adjustments must bemade:

The adjustments of the manual positioners, described in paragraphs 5.6,5.7, 5.8, 5.9, allow the correct adjustment of the beam path so that theimage of a plane wave on the camera is actually a point image, and thatthe condenser indeed forms a parallel beam at its exit. In particular,the adjustment described in paragraph 5.8. allows a first adjustment ofthe condenser position.

The adjustment described in paragraph 5.10. allows the number of motorsteps per pixel to be obtained, useful for controlling the beamdeflection mirror (109).

The level of the reference wave is adjusted as indicated in paragraph5.11.

The adjustment described in paragraph 5.12. makes it possible to obtainappropriate control voltages for the piezoelectric actuator.

The adjustment described in paragraph 5.13. allows the adjustment of thebeam attenuator and makes it possible to obtain the attenuation andphase shift constants characterizing it.

The adjustment described in paragraph 5.14. enables the constant K to beobtained.

The aperture of the diaphragms and the position of the semi-transparentmirror (116) must be adjusted so as to obtain a centered image withoutaliasing. This adjustment is described partly in paragraph 5.14 and iscompleted in paragraph 5.15.

The reference wave must be recorded, as described in paragraph 5.16.

5.4. Handling of filters

Adjustment operations require constant handling of filters to adjust theintensity received on the sensor. These handling operations are notsystematically described in the rest of the text.

The filter at (120) determines the intensity of the reference wave. Itsvalue is determined in a particular adjustment phase. Subsequently, whena reference wave is necessary, the filter thus determined is inserted.When the reference wave must be eliminated, an opaque element isinserted at (120).

The filter at (103) determines the intensity of the illuminating wave.Its value depends on the ongoing adjustment operations. For most of theoperations, the filter is adjusted so that the intensity received on thesensor (118) is high, without reaching saturation. For certainoperations, the sensor is saturated. For others, the illuminating wavemust be eliminated, which is done by inserting an opaque element.

The filter at (101) is used only for adjustment operations requiringvisual monitoring of the beam, identified by its diffusion disk on apiece of white paper. The filter is then adjusted so that the diffusiondisk is visible without being dangerous to the eye.

If only the illuminating wave is present, the term relative intensity ofthe image received on the CCD sensor will refer to the ratio of theintensity received on the sensor to the intensity at the output of thefilter (103). In an operation in which the aim is to maximize therelative intensity received on the sensor, it is necessary to changefilters regularly in order to keep the intensity at a level measurableby the sensor.

5.5. Programs currently used

Certain simple programs are frequently used during adjustment, withoutthis being indicated each time:

Displacement of the mirror (109): as this mirror is motor-driven, aprogram is needed in order to modify its position. This program asks theuser for a number of steps and an axis number corresponding either tothe motor (213) or to the motor (208), and then has this motor carry outthe number of steps requested.

Display of image received on the sensor: a program permits the directdisplay on the computer screen of the image received on the sensor(118).

Display of image and characteristics of the maximum: this program allowsthe direct display on the computer screen of the image received on thesensor (118). It also displays the maximum value detected by the sensor,the coordinates of the corresponding point, and the ratio between theintensity of this point and the sum of the intensities of its 8neighbors. This program is used to check the appearance of an image, tocheck the non-saturation of the sensor (maximum value lower than 255),to determined the coordinates and the value of the maximum, to evaluatethe point nature (punctuality) of this maximum by direct observation ofthe image and by the use of the displayed values: the (relative)intensity of the maximum must be as high as possible, as must also theratio of its intensity to that of its neighbors.

5.6. Adjustment of position of laser (100) and mirror (121)

In a first phase, the illuminating wave is eliminated and the positionof the sensor (100) is set so as to actually aim at the center of themirror (121), which is verified by following the path of the beam bymeans of a piece of paper enabling it to be displayed. The position ofthe mirror (121) is then adjusted so that the reference beam actuallypasses through the lens (123) and reaches the camera. The reference beammust be centered on the sensor (118).

5.7. Translation adjustment of camera position and adjustment of mirror(102)

The position of the camera is adjusted in translation by sending aparallel beam directly on to the microscope objective. For this purpose,elements (106)(108)(111)(105)(104) are temporarily removed, thereference wave is eliminated, and the microscope objective is placed ina position roughly focussed on the object. The object (112) used is atransparent plate and optical oil is placed between (112) and (113). Theangular position of (102) is then adjusted so that the beam reachesdirectly the center of the mirror (109). The position of the mirror(109) is adjusted so that the parallel beam enters directly into theobjective (113) and is fine-adjusted so as to maximize the relativeintensity of the signal received on the CCD sensor. The position of theCCD sensor is then adjusted in translation in the direction of theoptical axis so that the image produced is a perfect point image, thenadjusted in translation in the directions orthogonal to the optical axisso that this point is at the center of the useful area of the sensor. Itis finally re-adjusted in translation in the direction of the opticalaxis.

5.8. Condenser (111) position adjustment and obtaining the position ofthe optical center

The elements (106)(108)(111) are putback in place. Oil for themicroscope is placed between (111) and (112) and between (112) and(113). A piece of white cardboard is placed on the mirror (109) todiffuse the light. The diaphragm (107) is open to the maximum positionand the aperture of the diaphragm (114) is about 6 mm. The microscopeobjective is placed in a roughly focussed position. The position of thecondenser (111) is adjusted so as to obtain on the CCD a slightlygranular clear disc with the highest possible radius and with a roughlyconstant intensity over the entire disc. In FIG. 17, the useful area ofthe CCD (1200) is represented, and the clear disc (1201) stands outagainst this black background. A specific program is then used todetermine the coordinates C_(x), C_(y) of the optical center and theradius R of the disc. This program determines:

the lines (1206) and (1205) constituting the right and left limits ofthe disc (1201).

the line (1207) defined as the middle of the lines (1206) and (1205).

the lines (1202) and (1203) constituting the upper and lower limits ofthe disc (1201).

the line (1204) defined as the middle of the lines (1202) and (1203).

the optical center, which is the intersection of the lines (1204) and(1207)

the radius R of the disc, equal to half the distance between the lines(1205) and (1206).

5.9. Adjustment of lens (106) position

The piece of white cardboard placed on the mirror (109) is removed. Theposition of the mirror is modified in order to bring the illuminatedpoint on to the edge of the disc previously obtained. The position ofthe lens (106) is then adjusted along its axis so as to have the mostperfect possible point image on the CCD sensor.

5.10. Determining the number of steps per pixel

The piece of paper blocking the mirror is then removed. The motor ismoved along an axis by a known number of steps. The position in pixelsof the point of maximum intensity is noted before and after themovement, and the number of pixels covered is deduced therefrom. Theratio${{steps\_ per}{\_ pixel}} = \frac{{number}\quad {of}\quad {steps}}{{number}\quad {of}\quad {pixels}}$

is then calculated. The same is done on the other axis and the smallestratio obtained is used.

5.11. Reference wave level adjustment

If we consider a reference wave and an illuminating wave having the samemaximum intensity upon their arrival on the sensor, adding up inamplitude when they are in phase, the non-saturation requirement of thesensor is that the common amplitude of the two waves is half theamplitude saturating the sensor, or equivalently that the commonintensity of the two waves is one-fourth the intensity saturating thesensor. To adjust the level of the reference wave at this value, theilluminating wave is eliminated and the value of the filter (120) isadjusted to obtain an image whose maximum level is about one-fourth themaximum level authorized by the acquisition card, i.e. in the case of asampling on 8 bits of the video signal, a level of about 64. Beforeimaging, the maximum level of the illuminating wave must be adjusted inthe same manner.

5.12. Piezoelectric actuator control voltage adjustment

This calibration of the piezoelectric actuator can be performed outsidethe system using a known interferometric method. Three positions of theactuator are used. The movement of the mirror between each position mustbe $\frac{\lambda}{3\sqrt{2}}.$

The control voltages corresponding to each position must be determined,the actuator having a regular cycle to prevent hysteresis effects.

However, it is also possible to carry out this adjustment while theactuator is in place. This allows compensation for the inaccuracy on theorientation of the mirror and provides a simple calibration procedurenot requiring any specific equipment.

This is done by using a program which varies the control voltages andwhich is described by the algorithm of FIG. 10. Before starting theprogram, the position of the mirror (109) must be adjusted so that thepoint produced on the CCD sensor using an object consisting of acompletely transparent plate is at the center of the sensor. To utilizethis program, the object used must be highly diffusing. Use may be made,for example, of a piece of white paper soaked in gelatin and then placedbetween the plate and the cover glass. The paper must be sufficientlythick to stop the direct beam and sufficiently thin to allow the passageof a diffused beam. The diaphragm (107) is at its maximum aperture andthe diaphragm (114) is adjusted for an aperture of about 0.8 mm. Thereference wave then interferes on the CCD sensor with the clear discproduced by the wave coming from the object. The intensity of theilluminating wave must be adjusted so that the sensor is near itssaturation limit.

If the maximum voltage applied to the actuator is Umax, the voltagescorresponding to the phase shifts of${- \frac{2\quad \pi}{3}},0,\frac{2\quad \pi}{3}$

will be respectively Umax/2−diff_bas, Umax/2, Umax/2+diff_haut, wherediff_haut and diff_bas are chosen to produce the indicated phase shifts.To prevent any hysteresis effect, with each acquisition the voltage isinitialized to 0, the different images are acquired in the increasingorder of voltages applied to the actuator, and a final voltage Umax isfinally applied so that the same cycle is always used. This cycle mustalso be used in the normal operating phase. Imaging with the phaseshifts indicated above allows the calculation of a frequencyrepresentation by applying to each pixel P the formula${S\quad (P)} = \left\lbrack {{\frac{1}{6}\quad \left( {{2I\quad \left( {P,0} \right)} - {I\quad \left( {P,\frac{2\quad \pi}{3}} \right)} - {I\quad \left( {P,{- \frac{2\quad \pi}{3}}} \right)}} \right)} + {j\quad \frac{1}{2\sqrt{3}}\quad \left( {{I\quad \left( {P,\frac{2\quad \pi}{3}} \right)} - {I\quad \left( {P,{- \frac{2\quad \pi}{3}}} \right)}} \right)}} \right\rbrack$

in which the reference wave has been replaced by a constant and theexpression I(P,α) designates the intensity recorded at the point P for aphase shift α.

The adjustment of the actuator control voltages consists in evaluatingdiff_haut and diff_bas. The principle is to obtain two frequencyrepresentations shifted by π/3 in relation to each other. The secondimage can be phase corrected by multiplying it by e^(−jπ/3) and theaverage difference between the first image and this phase-correctedimage can be calculated. This difference is minimum when the voltagesdiff_haut and diff_bas are correctly adjusted. The phase shifts allowingthe acquisition of the second elementary frequency image are thus${{- \frac{2\quad \pi}{3}} + \frac{\quad \pi}{3}},\frac{\quad \pi}{3},{\frac{\quad {2\quad \pi}}{3} + \frac{\pi}{3}}$

and correspond respectively, as a first approximation, to the voltagesUmax/2−dif_bas+diff_bas/2, Umax/2+(diff_bas+diff_haut)/2,Umax/2+diff_haut+diff_haut/2. The adjustment program calculates theaverage difference for series of values of diff_haut and diff_bas andchooses those which correspond to the minimum average difference. Itsalgorithm is shown in detail in FIG. 10.

The basic steps of this algorithm are:

(600): Acquisition of images. The acquisition procedure is depicted inFIG. 11. By “images” is meant here six interference patterns receivedconsecutively by the CCD sensor. This procedure always carries out thesame cycle beginning with the voltage 0 and ending with the voltageUmax. The waiting time after the application of a zero voltage to thepiezoelectric enables the latter to become stabilized. The waiting timebefore the start of the acquisition avoids the acquisition of a framewhich may have been exposed before the application of the desiredexposure conditions. The end of the exposure of an image is indicated onthe acquisition cards by the start of the transfer of the correspondingimage. The use of an exposure time shorter than the image transfer timeprevents the image from being affected by transient states. Giving theprocess maximum priority prevents the disturbance of acquisition byother system tasks under a multitask operating system. The six imagesare acquired successively and in real time (no image lost) by theacquisition card in order to minimize the time during which vibrationscan affect the result. Each image has the horizontal dimension of hpixand vertical dimension of vpix. During acquisition, images aretransferred automatically by the acquisition card into an array which isreserved for them in the computer's central memory. Following theacquisition procedure, there is a resulting array I[a,b,i,j], the indexa corresponding to the phase difference, the index b corresponding tothe image (phase-shifted or not phase-shifted), the indices i and jbeing the coordinates in pixels and varying respectively from 0 tohpix−1 and from 0 to vpix−1.

(601): Calculation of the two frequency representations. These arestored in the arrays of complexes S0[i,j] and S1[i,j], applying theformulas:${{S0}\left\lbrack {i,j} \right\rbrack} = {{\frac{1}{6}\quad \left( {{2{I\left\lbrack {0,0,i,j} \right\rbrack}} - {I\left\lbrack {1,0,i,j} \right\rbrack} - {I\left\lbrack {2,0,i,j} \right\rbrack}} \right)} + {\overset{\sim}{j}\quad \frac{1}{2\sqrt{3}}\quad \left( {{I\left\lbrack {1,0,i,j} \right\rbrack} - {I\left\lbrack {2,0,i,j} \right\rbrack}} \right)}}$${{S1}\left\lbrack {i,j} \right\rbrack} = {{\frac{1}{6}\quad \left( {{2{I\left\lbrack {0,1,i,j} \right\rbrack}} - {I\left\lbrack {1,1,i,j} \right\rbrack} - {I\left\lbrack {2,1,i,j} \right\rbrack}} \right)} + {\overset{\sim}{j}\quad \frac{1}{2\sqrt{3}}\quad \left( {{I\left\lbrack {1,1,i,j} \right\rbrack} - {I\left\lbrack {2,1,i,j} \right\rbrack}} \right)}}$

(602): The program modifies the array S1 by multiplying each of itselements by e^(−jπ/3).

(603): The program calculates the maximum value${mod\_ max} = {\max\limits_{\underset{0 \leq j \leq {{vpix} - 1}}{0 \leq j \leq {{hpix} - 1}}}\quad {{{S0}\left\lbrack {i,j} \right\rbrack}}}$

of the modulus on S0.

(604): The program calculates the standard difference between the twoarrays as follows:

The program initializes ecart and nombre_valeurs to 0 and goes throughall the points i,j by testing the condition (605): |S0[i,j]|≧0,5mod_max. Each time this condition is met, it carries out the followingoperations (606):

ecart+=|S 0[i,j]−S 1[i,j]|

nombre_valeurs+=1

When the program has gone through the indices (i,j) it divides the ecartby nombre valeurs (607), giving it the average difference. Thisdifference consequently integrates only values for which theilluminating wave is sufficiently strong, in order to prevent too noisya result. It is stored in an array.

(608): The program goes on to the next value of the pair(diff_haut,diff_bas) and reiterates the operations until the series ofdifferences has been calculated.

(609): The program applies a low-pass filter to the array obtained inorder to limit the noise due to system vibrations. The low-pass filterused is represented in the frequency domain by a step going from a valueof 1 at low frequencies to a value of 0 at high frequencies, and itsbandwidth is determined empirically to have a good limitation of noisewithout excessively deforming the curve.

(610): The array is represented graphically in order to check its shape.

(611): The value of the pair (diff_bas, diff_haut) corresponding to theminimum is shown.

The series of values of the pair (diff_bas,diff_haut) are determined asfollows:

In a first stage, the program varies the values of diff_haut anddiff_bas, leaving them equal to each other. For example, they may varybetween 0 and $\frac{Umax}{4}$

in steps of $\frac{Umax}{4096}$

if use is made of 12 conversion bits. The result is an array of 1024elements (the differences calculated for each value of(diff_haut,diff_bas) ) that the program filters in order to eliminatethe noise (609) and which it represents graphically (610). The indexvalue of the array corresponding to the minimum difference thencorresponds to the correct value of the pair (diff_bas,diff_haut) andthis pair is displayed (611).

In a second stage, the program is restarted by setting diff_bas to thevalue previously obtained and by varying only diff_haut. A new minimumis thus obtained, which corresponds to a more precise value ofdiff_haut.

In a third stage, the program is restarted by setting diff_haut to thevalue previously obtained and by varying only diff_bas, obtaining a moreprecise value of diff_bas.

The operator can reiterate these steps by varying separately andalternately diff_bas and diff_haut, but the maximum accuracy on thesevalues is obtained quite rapidly.

5.13. Adjustment of polarizer (105) and of polarization rotator (104)

The following step consists in adjusting the position of thepolarization rotator (104) and of the polarizer (105). This device isdesigned to provide a controlled attenuation of the illuminating beam bycontrolling the phase rotator, and the assembly will be called the“optical switch”. It has a closed position corresponding to a lowintensity through it, and an open position corresponding to a higherintensity.

The object used is the same as in the preceding step, and the adjustmentof the mirror (109) and of the diaphragms is also the same. Thereference wave is first eliminated. In a first stage, the polarizer isput in place and adjusted to maximize the intensity going through it andreceived on the sensor (118). In a second stage, the polarizationrotator is put in place. The voltage corresponding to a statearbitrarily defined as closed is applied to it and it is positioned inrotation to minimize the intensity going through the entire switch. In athird stage, the reference wave is re-established and a program is usedwhich calculates the ratio of the intensities and the phase differencebetween the two positions of the polarization rotator, a ratio whichwill be needed during the image acquisition phases in order to combinethe waves corresponding to the open and closed states of the switch. Thealgorithm of this program is shown in FIG. 12. The stages are thefollowing:

(800): The program first acquires the images using the acquiert_imagesprocedure described in FIG. 13. By “images” is meant here sixinterference patterns received consecutively by the CCD sensor. Thisprocedure always performs the same cycle beginning with the voltage 0and ending with the voltage Umax. The waiting time after the applicationof a zero voltage to the piezoelectric allows the latter to becomestabilized. The waiting time before the start of the acquisitionprevents the acquisition of a frame which may have been exposed beforethe application of the desired exposure conditions. The end of theexposure of an image is indicated on the acquisition cards by the startof the transfer of the corresponding image. The use of an exposure timeshorter than the image transfer time prevents the image from beingaffected by transient conditions. Giving the process maximum priorityprevents the disturbance of the acquisition by other system tasks undera multitask operating system. The six images are acquired successivelyand in real time (no image lost) by the acquisition card in order tominimize the time during which vibrations may affect the result. Eachimage has the horizontal dimension of hpix and vertical dimension ofvpix. During the acquisition, the images are transferred automaticallyby the acquisition card into the array reserved for them in thecomputer's central memory.

(801): The program calculates the two frequency representations S0 andS1 which differ by the state of the switch. S0 corresponds to the openswitch and S1 to the closed switch.${{S0}\left\lbrack {i,j} \right\rbrack} = {{\frac{1}{6}\quad \left( {{2{I\left\lbrack {0,0,i,j} \right\rbrack}} - {I\left\lbrack {1,0,i,j} \right\rbrack} - {I\left\lbrack {2,0,i,j} \right\rbrack}} \right)} + {\overset{\sim}{j}\quad \frac{1}{2\sqrt{3}}\quad \left( {{I\left\lbrack {1,0,i,j} \right\rbrack} - {I\left\lbrack {2,0,i,j} \right\rbrack}} \right)}}$${{S1}\left\lbrack {i,j} \right\rbrack} = {{\frac{1}{6}\quad \left( {{2{I\left\lbrack {0,1,i,j} \right\rbrack}} - {I\left\lbrack {1,1,i,j} \right\rbrack} - {I\left\lbrack {2,1,i,j} \right\rbrack}} \right)} + {\overset{\sim}{j}\quad \frac{1}{2\sqrt{3}}\quad \left( {{I\left\lbrack {1,1,i,j} \right\rbrack} - {I\left\lbrack {2,1,i,j} \right\rbrack}} \right)}}$

(802): The program calculates the maximum value${mod\_ max} = {\max\limits_{\underset{0 \leq j \leq {{vpix} - 1}}{0 \leq j \leq {{hpix} - 1}}}\quad {{{S0}\left\lbrack {i,j} \right\rbrack}}}$

reached by the modulus of the elements of array S0.

(803): The program calculates the average ratio between the twofrequency representations. It initializes rapport and nombre_valeurs to0 then goes through all the indices (i,j) while testing the condition(804): |S0[i,j]|≧0,5 mod_max

When the condition has been checked, it carries out (805):${rapport}+=\frac{{S0}\left\lbrack {i,j} \right\rbrack}{{S1}\left\lbrack {i,j} \right\rbrack}$

 nombre_valeurs+=1

When all the indices i,j have been gone through, the program dividesrapport by nombre_valeurs (806) which gives the sought ratio.

(807): The program calculates the average rapport_moy of the values ofrapport obtained since it started.

(808): The program displays the real and imaginary parts as well as themodulus of rapport and rapport_moy.

(809): The program reiterates this procedure continuously to allowcontinuous adjustment. The program is ended on an instruction from theoperator.

The angular position of the polarization rotator must be adjusted sothat the modulus of rapport is roughly equal to 30. The program is thenstopped and restarted and, after a sufficient number of iterations, theaverage complex value rapport_moy is noted and will serve as the basisfor the following operations.

5.14. Obtaining the constant K and adjusting the diaghram (114) and themirror (116)

K is the maximum value in pixels corresponding to the maximum spatialfrequency of the wave where 1/λ_(v) where λ_(v) is the wavelength in theobserved medium, assumed to have an index equal to the nominal indexn_(v) of the objective. The nominal index of the objective is the indexfor which it was designed and for which it creates no sphericalaberration. It is also the index of the optical oil to be used with theobjective.

There are K pixels between the frequencies 0 and 1/λ_(v). The frequencystep along an axis is thus $\frac{1}{K\quad \lambda_{v}}.$

The frequencies vary in all from −1/λ_(v) to 1/λ_(v) in steps of$\frac{1}{K\quad \lambda_{v}}.$

If N is the total number of pixels along each axis taken into accountfor the Fourier transform, N values of frequencies are taken intoaccount, from $- \frac{N}{2\quad K\quad \lambda_{v}}$

to $\frac{N}{2\quad K\quad \lambda_{v}}.$

After transformation, N position values are obtained with a positionstep equal to half the inverse of the maximum frequency beforetransformation.

The position step is thus$\frac{1}{\frac{N}{2\frac{N}{2K\quad \lambda_{v}}}} = {\frac{K\quad \lambda_{v}}{N}.}$

If we consider two points between which the distance in pixels isD_(pix) and the real distance is D_(reel), we have:$D_{reel} = {\frac{K\quad \lambda_{v}}{N}D_{pix}}$

so that $K = {\frac{N}{\lambda_{v}}{\frac{D_{reel}}{D_{pix}}.}}$

The wavelength to be considered here is the wavelength in the material,assumed to have an index equal to the nominal index n_(v) of theobjective or: $\lambda_{v} = {\frac{\lambda}{n_{v}}.}$

We finally have:$K = {\frac{n_{v}}{\lambda}\frac{N}{D_{pix}}{D_{reel}.}}$

To obtain the constant K, we produce the image of an objectivemicrometer for which the real distances are known, and then apply theabove formula.

This is done using a focussing program which will be re-used laterwhenever focussing on a sample is necessary before three-dimensionalimaging. The algorithm of this program is shown in FIG. 14. Its mainsteps are:

(1000): The program acquires an image using the acquiert_imagesprocedure of FIG. 13. This yields an array of integers (unsigned chartype for 8 bits) I[p,c,i,j] where the index p varying from 0 to 2corresponds to the phase state, the index c varying from 0 to 1corresponds to the state of the switch (0=open, 1=closed) and indices iand j varying from 0 to hpix−1 and from 0 to vpix−1 correspond to thecoordinates of the pixel.

(1001): An array of Booleans H[i,j] is generated: it is initialized to 0then, for each pixel, the maximum value reached by I[p,0,i,j] on thethree images corresponding to the open position of the switch iscalculated. If this value is equal to 255 (highest value of thedigitizer), the images corresponding to the closed position of theswitch must be used in calculating the frequency associated with thepixel and with its 8 immediate neighbors, and the array H[i,j] is set to1 for these 9 pixels.

(1002): The frequency representation S[i,j] of complex numbers isgenerated: for each pixel, the value is generated according to thefollowing equations:$\left. {{S\left\lbrack {i,j} \right\rbrack} = \left\lbrack {{\frac{1}{6}\left( {{2{I\left\lbrack {0,{H\left\lbrack {i,j} \right\rbrack},i,j} \right\rbrack}} - {I\left\lbrack {1,{H\left\lbrack {i,j} \right\rbrack},i,j} \right\rbrack} - {I\left\lbrack {2,{H\left\lbrack {i,j} \right\rbrack},i,j} \right\rbrack}} \right)} + {\overset{\sim}{j}\frac{1}{2\sqrt{3}}\left( {{I\left\lbrack {1,{H\left\lbrack {i,j} \right\rbrack},i,j} \right\rbrack} - {I\left\lbrack {2,{H\left\lbrack {i,j} \right\rbrack},i,j} \right\rbrack}} \right)}}\quad \right)} \right\rbrack \quad \left( {1 + {\left( {{rapport\_ moy} - 1} \right){H\left\lbrack {i,j} \right\rbrack}}} \right)$

If H[i,j] is equal to 1, the complex value thus obtained is consequentlymultiplied by the complex number rapport_moy obtained during the switchcalibration operation in order to give the final value of the element ofarray S[i,j], in order to take into account the phase shift andabsorption induced by the switch.

(1003): The program limits the array S to dimensions of 512×512. Theprogram then carries out optionally one or the other, or none, of thefollowing two operations:

averaging over a width of 2, which brings the array S to an array S′ ofdimensions 256×256 withS′[i,j]=S[2i,2j]+S[2i+1,2j]+S[2i,2j+1]+S[2i+1,2j+1]. This averaging,coupled with a reduction in the aperture of the diaphragm (114), makesit possible to reduce the diameter of the observed area and to reducethe calculation time. It is equivalent to a low-pass filter followed bysub-sampling.

limitation of all the observed frequencies to a square of 256×256 pixelswith S″[i,j]=S[128+i,128+j]. This allows a reduction in the calculationtime at the cost of a reduction in resolution. However, in the presentcase, the program carries out neither of these two operations.

(1004): The program then performs the inverse Fourier transform of thearray thus obtained.

(1005): It displays the results on the screen, extracting the modulus,the real part or the imaginary part. In the presence case, it willdisplay the modulus. Whatever the variable displayed, the correspondingarray of real numbers is first normalized either in relation to theaverage value or in relation to the maximum value. The program alsowrites on the disk the corresponding file of real values. When the realpart or the imaginary part are represented, it is essential that thepoint of impact of the direct beam should be at (256,256) on the imageof size 512×512, otherwise a modulation becomes visible. When themodulus is represented, the exact point of impact of the direct beamdoes not affect the result signficantly.

(1006): The program starts again during the acquisition of a new image,thus operating continuously. It stops on an instruction from theoperator.

To obtain the image of the micrometer, the objective is first placed ina roughly focussed position, the micrometer having been introduced as anobject. In a first step, the reference wave is eliminated, thediaphragms are at maximum aperture, and the program for the directreal-time display of the image received on the sensor is started, thefilters at (103) and the polarization rotator being set to allow thepassage of sufficient intensity to saturate significantly the sensor atthe direct point of impact of the beam. The object is then moved in thehorizontal plane by means of the corresponding positioner until theimage made up of many aligned intense points characteristic of themicrometer appears. The micrometer is then correctly positioned underthe objective.

The diaphragms at (107) and (114) are then set for an aperture of about8 mm. The filters at (103) are then set so that the maximum intensity onthe CCD is at a level of about a fourth of the maximum value of thedigitizer, or 256/4=64. The reference wave is re-introduced. Thefocussing program is then started. The diaphragm (114) must be adjustedso that it is clearly visible on the displayed image, while being asopen as possible. If the image is not correctly centered, it is possibleto improve its centering either by modifying the orientation of themirror (116), in which case it may be necessary to readjust theorientation of the mirror (121), or by modifying the position of thediaphragm (114). The diaphragm at (107) must be adjusted so that theobserved area appears uniformly illuminated. The focussing program isthen stopped, the reference wave is eliminated and the intensity of thebeam is readjusted as previously. The reference wave is thenre-introduced and the focussing program restarted.

The microscope objective is then moved by the focussing device so as toobtain a good image of the micrometer. To facilitate focusing, it isadvisable to display the part of the micrometer where lines of differentlength are present. This limits “false focussing” due to interferencephenomena ahead of the micrometer. Between two movements it is necessaryto release the manual focussing device to obtain an image not disturbedby the vibrations. When a good image has been obtained, the program isstopped and the obtained image is used to obtain the distance in numberof pixels between two lines, the metric distance between these linesbeing known. If the distance between two graduations separated byD_(real) micrometers is the image thus obtained of D_(pix) pixels, ifthe nominal index of the objective is n_(v) (in general, n_(v)=1.5) andif the wavelength of the laser in vacuum is λ (λ=0.633 micrometers) andif the number of points of the Fourier transform is N (N=512) then wehave: ${K = {\frac{n_{v}}{\lambda}\frac{N}{D_{pix}}D_{real}}},$

where of course D_(real) and λ are in the same unit.

5.15. Diaphraom adjustment

The three-dimensional image to be calculated has a side measuring 256pixels, which allows the file sizes and calculation times to be limited.The adjustment of the diaphragm consists in re-using the focussingprogram, this time with the intermediate averaging option, and adjustingthe diaphragm (114) so that its image is clearly visible while being aslarge as possible. The diaphragm (107) is then adjusted so that it isslightly more open than the minimum allowing regular illumination of theobserved part of the sample.

5.16. Reference wave recording

Knowing the reference wave is essential for the precise calculation ofthe complex values of the wave reaching the sensor. It must be recordedindependently of the constant average noise value that characterizeseach pixel. A specific program is used for this purpose. In a firststep, the illuminating and reference waves are eliminated and theprogram records the optically dark background which results on the CCDsensor. It averages the intensity obtained on 100 images to have anoise-free optically dark background. In a second step, the referencewave is re-established and the illuminating wave remains eliminated. Theprogram records the resulting image, the average on 100 images for noiseremoval. The program then calculates the difference between the image ofthe reference wave alone and the background image, and records theresulting image in an array Iref[i,j] where i varies from 0 to hpix−1and j varies from 0 to vpix−1.

5.17. Focussing on the studied object

This stage must be reiterated for each sample for which an image isdesired. The illuminating wave is reestablished. The sample to bestudied is put in place. The mirror (109) is adjusted so that the directimpact point of the illuminating beam is at the center of the sensor.The filters at (103) are adjusted so that, in the absence of a referencewave, the maximum intensity received on the CCD sensor is about 64. Thereference wave is then re-established. The focussing program is startedwith the intermediate averaging option and the position of the objectiveis adjusted by means of the focussing device to obtain a clear image ofthe region of interest of the sample.

5.18. Adjustment of condenser position and setting of diaphragm (107)

After the focussing phase, the position of the condenser must beadjusted so that the image, in the sample, of the illuminated part ofthe mirror coincides with the observed part of the sample. The referencewave is removed, the illuminating beam attenuation device is placed inthe open position, the filters located along the path of theilluminating beam are removed, and a wet piece of white paper is placedon the surface of the mirror (109) so as to form a diffusing surface ofsmall thickness that may be considered assimilated with the surface ofthe mirror. A specific program for the display of the image received onthe CCD is used. This program averages the image, for example over eightsuccessive acquisitions, and represents, on the screen, the root of thevalue obtained at each point so as to make the image more visible evenin the absence of a high brightness. The diaphragm (107) is first openedwidely, and the position of the condenser is adjusted to obtain ahomogeneous clear disc of long radius. It is then closed again graduallyup to the image visibility limit, the position of the condenser beingitself readjusted gradually. The final aperture of the diaphragm (107)must be clearly smaller than that determined in 5.15.

The filters are then re-introduced, the reference wave is alsore-introduced, and the focussing program is started. The image of thediaphragm (114) should be seen with a central light spot correspondingto the out-of-focus (hence blurred) image of the diaphragm (107). Theposition of the condenser can then be adjusted in a plane orthogonal tothe optical axis so that the central light spot is centered.

The diaphragm (107) is then adjusted so as to be slightly more open thanthe minimum allowing regular illumination of the observed area.

5.19. Filter adjustment

The filters at (103) are adjusted so that, in the absence of a referencewave and in the open position of the beam attenuator, the maximumintensity measured on the CCD sensor is about 64. It can also be useful,just before acquisition, to again adjust the position of thepolarization rotator (104) and of the polarizer (101). This adjustmentcan be carried out using the same program as in 5.13, but leaving theobserved sample in place and without modifying the adjustment of thediaphragms and filters. This adjustment allows compensation for thetime-wise drift of polarization rotator characteristics.

5.20. Acquisition stare

This stage allows the acquisition of two-dimensional frequencyrepresentations from which the three-dimensional representation will becalculated. The point of direct impact of the illuminating wave is movedalong a series of radii starting at the optical center and having alength Rb slightly smaller than R, for example Rb=R−6, the opticalcenter and the radius R having been determined in stage 5.8. nbangleswill be used to denote the total number of radii that will be traversed,for example nbangles=9. nbim is the number of two-dimensional frequencyrepresentations to be acquired and its value is${nbim} = {\frac{Rb}{2}{{nbangles}.}}$

At each stage, the program calculates and stores in a file fich_acquis afrequency representation which will be used in the three-dimensionalcalculation phase. The size of the image acquired from the camera ishpix×vpix, but this size is divided by two to obtain frequencyrepresentations of dimension hel×vel, according to the intermediateaveraging principle already used in the focussing program.

The acquisition program is represented in detail by the algorithm ofFIG. 18 whose stages are the following:

(1300): The basic data of the acquisition are stored in filefich_acquis:

nbim: number of two-dimensional frequency representations

hel=hpix/2: final number of points in the horizontal direction

vel=vpix/2: final number of points in the vertical direction

(1301): Waiting time allowing the absorption of vibrations created bythe movement of the positioner (110). A waiting time of about 2 s can besuitable.

(1302): The program acquires the images through the acquiert_imagesprocedure of FIG. 13. One thus obtains an array of integers (unsignedchar type for 8 bits) I[p,c,i,j] where the index p varying from 0 to 2corresponds to the phase condition, the index c varying from 0 to 1corresponds to the condition of the switch (0=open, 1=closed) and theindices i and j varying respectively from 0 to hpix−1 and from 0 tovpix−1 correspond to the coordinates of the pixel.

(1303): An array of Booleans H[i,j] is generated: it is initialized to0, then, for each pixel, the maximum value reached on the three imagescorresponding to the open position of the switch is calculated. If thisvalue is equal to 255 (highest value of the digitizer), the imagescorresponding to the closed position of the switch must be used in thecalculation of the frequency associated with the pixel and with its 8immediate neighbors, and the array H is set to 1 for these 9 pixels.

(1304): Calculation of rapport_moy. The calculation of rapport_moy,already carried out during the switch calibration operation, must beperformed again with each image so as to compensate for time drifts. Theprogram first determines:${Vmax} = {\max\limits_{{({i,j})}\varepsilon \quad {EI}}{{{\frac{1}{6\sqrt{{Iref}\left\lbrack {i,j} \right\rbrack}}\left( {{2{I\left\lbrack {0,0,i,j} \right\rbrack}} - {I\left\lbrack {1,0,i,j} \right\rbrack} - {I\left\lbrack {2,0,i,j} \right\rbrack}} \right)} + {\overset{\sim}{j}\frac{1}{2\sqrt{3{{Iref}\left\lbrack {i,j} \right\rbrack}}}\left( {{I\left\lbrack {1,0,i,j} \right\rbrack} - {I\left\lbrack {2,0,i,j} \right\rbrack}} \right)}}}}$

where E1 is the set of pairs (i,j) verifying H[i,j]=0

The program then calculates:${rapport\_ moy} = {\frac{1}{N2}{\sum\limits_{{({i,j})} \in {E2}}^{\quad}\quad {\frac{{\frac{1}{6}\left( {{2{I\left\lbrack {0,0,i,j} \right\rbrack}} - {I\left\lbrack {1,0,i,j} \right\rbrack} - {I\left\lbrack {2,0,i,j} \right\rbrack}} \right)} + {\overset{\sim}{j}\frac{1}{2\sqrt{3}}\left( \left. {{I\left\lbrack {1,0,i,j} \right\rbrack} - {I\left( {2,0,i,j} \right.}} \right\rbrack \right)}}{{\frac{1}{6}\left( {{2{I\left\lbrack {0,1,i,j} \right\rbrack}} - {I\left\lbrack {1,1,i,j} \right\rbrack} - {I\left\lbrack {2,1,i,j} \right\rbrack}} \right)} + {\overset{\sim}{j}\frac{1}{2\sqrt{3}}\left( {{I\left\lbrack {I,1,i,j} \right\rbrack} - {I\left( {2,1,i,j} \right\rbrack}} \right)}}}}}$

E2 is the set of pairs (i,j) verifying H[i,j]=0 and${{{\frac{1}{6\sqrt{{Iref}\left\lbrack {i,j} \right\rbrack}}\left( {{2{I\left\lbrack {0,0,i,j} \right\rbrack}} - {I\left\lbrack {1,0,i,j} \right\rbrack} - {I\left\lbrack {2,0,i,j} \right\rbrack}} \right)} + {\overset{\sim}{j}\frac{1}{2\sqrt{3{{Iref}\left\lbrack {i,j} \right\rbrack}}}\left( {I\left\lbrack {1,0,i,j} \right\rbrack} \right)}}} \geq {{coef} \cdot {Vmax}}$

with, for example, coef=0.5. The coefficient coef can be adjusted sothat E2 contains a sufficient number of points. N2 is the number ofelements of E2.

(1305): The frequency representation S[i,j] of complex numbers isgenerated: for each pixel, the value is generated according to thefollowing equations:${S\left\lbrack {i,j} \right\rbrack} = {\left\lbrack {{\frac{1}{6\sqrt{{Iref}\left\lbrack {i,j} \right\rbrack}}\left( {{2\quad {I\left\lbrack {0,{H\left\lbrack {i,j} \right\rbrack},i,j} \right\rbrack}} - {I\left\lbrack {1,{H\left\lbrack {i,j} \right\rbrack},i,j} \right\rbrack} - {I\left\lbrack {2,{H\left\lbrack {i,j} \right\rbrack},i,j} \right\rbrack}} \right)} + {\overset{\sim}{j}\frac{1}{2\sqrt{3{{Iref}\left\lbrack {i,j} \right\rbrack}}}\left( {{I\left\lbrack {1,{H\left\lbrack {i,j} \right\rbrack},i,j} \right\rbrack} - {I\left\lbrack {2,{H\left\lbrack {i,j} \right\rbrack},i,j} \right\rbrack}} \right)}} \right\rbrack \left( {1 + {\left( {{rapport\_ moy} - 1} \right){H\left\lbrack {i,j} \right\rbrack}}} \right)}$

If H[i,j] is equal to 1, the complex value thus obtained is multipliedby the complex number rapport_moy obtained during the switch calibrationoperation to give the final value of the element of array S[i,j], inorder to take into account the phase shift and absorption induced by theswitch.

(1306): The point corresponding to the modulus maximum value of array Sis determined. Its coordinates (i_max,j_max) are recorded.

(1307): The “saught” position is calculated, the angle having the valuek.pas:${i\_ obj} = {G_{x} + {{\left( {k\quad \% \quad \left( \frac{Rb}{2} \right)} \right) \cdot \cos}\quad \left( {E\quad \left( {\frac{2\quad k}{Rb}\frac{2\quad \pi}{nbangles}} \right)} \right)}}$${j\_ obj} = {G_{x} + {{\left( {k\quad \% \quad \left( \frac{Rb}{2} \right)} \right) \cdot \sin}\quad \left( {E\quad \left( {\frac{2\quad k}{Rb}\frac{2\quad \pi}{nbangles}} \right)} \right)}}$

where i_obj and j_obj are coordinates of the objective position inpixels, and where the sign % designates the modulo, and where E(x)designates x rounded to the nearest integer.

The movement of the motors is calculated:

pas_(—) i=(i_obj−i_max).pas_par_pixel

pas_(—) j=(i_obj−j_max).pas_par_pixel

where pas_par_pixel is the number of steps of the motors per pixel ofmovement, determined experimentally during the adjustments.

(1308): One of the motors allows the movement of the direct impact pointof the beam in the direction of the axis i. It must complete a number ofsteps pas_i in the direction corresponding to an index i increasing forthe position of the direct impact point of the illuminating beam (forpas_i<0, it must complete a number of steps −pas_i in the oppositedirection). The other motor allows the movement of the direct impactpoint of the beam in the direction of the axis j. Similarly, it mustcomplete a number of steps pas_j.

(1309): An array of averaged frequencies is generated:

Each dimension of the initial array S is divided by two to give an arrayM_(k) with${M_{k}\left\lbrack {i,j} \right\rbrack} = {\sum\limits_{\substack{0 \leq p \leq 1 \\ 0 \leq q \leq 1}}\quad {S\left\lbrack {{{2\quad i} + p},{{2\quad j} + q}} \right\rbrack}}$

(1310): The point corresponding to the modulus maximum value of arrayM_(k) is determined, and its coordinates imax_(k), jmax_(k) and thevalue of this point max_moy=M_(k)[imax_(k), jmax_(k)] are recorded.

(1311): The elements of array M_(k) are normalized by dividing them bymax_moy:${M_{k}\left\lbrack {i,j} \right\rbrack} = \frac{M_{k}\left\lbrack {i,j} \right\rbrack}{max\_ moy}$

(1312): imax_(k), jmax_(k) and the frequency representation M_(k) arestored in file fich_acquis.

(1313): The algorithm ends when the angle is equal to 2π. The motor thenreturns to its initial position and the acquisition file fich_acquis isclosed.

5.21. Three-dimensional calculation

The acquisition procedure generated two-dimensional frequencyrepresentations M_(k), where the index k represents the order number ofeach representation. These representations have been averaged over awidth of 2, so that the values C_(x), C_(y) and K must be divided by 2to correspond to the new system of coordinates. These divided values arethe ones used below. The set of two-dimensional representations may beconsidered to be a representation with three dimensions (i,j,k) in whichthe index k represents the image index and the indices i and j representthe Cartesian coordinates of each representation. In the k-thtwo-dimensional frequency representation:

The relation between the coordinates and the characteristic frequencyis:

Kλf _(c) =[i−C _(x) ,j−C _(y) ,{square root over(K²−(i−C_(x))²−(j−C_(y))²])}

The relation between the coordinates and the illuminating frequency is:

Kλf _(e) =[imax_(k) −C _(x) ,jmax_(k) −C _(y) ,{square root over(K²−(imax_(k)−C_(x))²−(jmax_(k)−C_(y))²])}

The relation between the coordinates and the total frequency is thus:

Kλf _(i)=[i−imax_(k) ,j−max_(k),{square root over(K²−(i−C_(x))²−(j−C_(y))²)}−{square root over (K²−(imax_(k)−C_(x))²−(jmax_(k)−C_(y))²])}

The three-dimensional calculation procedure consists in principle ingenerating during a first stage a three-dimensional representation inthe form of an array F of dimensions fdim×fdim×fdim where fdim=512, thenin taking the Fourier transform to obtain an array U of the samedimensions corresponding to the representation u(r), and in which theindices thus correspond to the position vector. The representation F isan array in which the indices represent the coordinates of Kλf_(T), thezero being brought by translation to the point of coordinates(fdim/2,fdim/2,fdim/2). One thus obtains, from each point of coordinatesi,j of a two-dimensional representation M_(k), a point of thethree-dimensional representation by:

F[ni,nj,nk]=M _(k) [i,j]

where: ${ni} = {i - {imax}_{k} + \frac{fdim}{2}}$${nj} = {j - {jmax}_{k} + \frac{fdim}{2}}$${nk} = {\sqrt{K^{2} - \left( {i - C_{x}} \right)^{2} - \left( {j - C_{y}} \right)^{2}} - \sqrt{K^{2} - \left( {{imax}_{k} - C_{x}} \right)^{2} - \left( {{jmax}_{k} - C_{y}} \right)^{2}} + \frac{fdim}{2}}$

When a point of array F, of coordinates (ni,nj,nk), is obtainedsuccessively from several distinct two-dimensional representations, thevalue of F taken at this point is the average of the values obtainedfrom each of the two-dimensional representations. When this point isnever obtained, a zero value is assigned.

When the array F has been generated, the array U can be obtained bymeans of the inverse three-dimensional Fourier transform.

This method can be applied directly if the program has a sufficientrandom-access memory. The program whose algorithm is described in FIG.19 however makes it possible to carry out the calculations on a systemwhose RAM is limited, the files being stored on a sequential-accessmedium (hard disk).

For practical reasons, part of the Fourier transformation will becarried out as the array F is being generated, so that it isconsequently never really generated.

The program works in five stages. Each stage uses an input file storedon the hard disk of the computer and generates an output file alsostored on the hard disk, whose name is italicized in the figure. Theprogram could theoretically carry out more directly the operationsnecessary for the generation of the three-dimensional image, but thesize of the files involved is too large to enable them to be containedentirely in a computer memory. It is thus necessary to manage theirstorage on disk. As the reading/writing of file elements on disk takesplace faster if the elements are stored contiguously, the program mustbe designed to read and write on the disk only blocks of a sufficientsize. This is what the described algorithm renders possible, whereas adirect method would require reading/writing to non-contiguous addressesand would not be practicable owing to the time lost in disk accesses.The stages of the algorithm are the following:

(1400): Centering of two-dimensional frequency representations:

This procedure consists in translating the two-dimensionalrepresentations to go from a representation in the system of coordinates(i,j,k) to a representation in the system (ni,nj,k) whereni=i−imax+fdim/2,nj=j−jmax+fdim/2, the dimensions following ni and nj inthe array thus generated being fdim×fdim. The algorithm of this part isrepresented in FIG. 21. The stages are the following:

(1600): The values of hel,vel,nbim are read in the file fich_acquis.

(1601): The frequency representation M_(k) corresponding to the index kis transferred to central memory with the corresponding values imax_(k)and jmax_(k).

(1602): A translated frequency representation T_(k) of dimensionsfdim×fdim is generated with:

.T _(k) [ni,nj]=M _(k) [ni−fdim/2+imax_(k) ,nj−fdim/2+jmax_(k)]

when

0≦ni−fdim/ 2+imax_(k) ≦hel−1 and 0≦nj−fdim/2+jmax_(k) ≦hel−1.

.T_(k)[ni,nj]=0 in the other cases.

FIG. 20 shows an image of an origin with its point of coordinates(imax_(k), jmax_(k)) and an arbitrary drawing around this point, and thenew translated image.

(1603): The values of imax_(k), jmax_(k), and the representation T_(k)are stored in the file fich_centréin the following order:

imax_(k),jmax_(k),T_(k)[0,0],T_(k)[1,0], . . . T_(k)[fdim−1,0],

imax_(k),jmax_(k),T_(k)[0,1],T_(k)[1,1], . . . T_(k)[fdim−1,1], . . .

imax_(k),jmax_(k),T_(k)[0,fdim]T_(k)[1,fdim], . . .T_(k)[fdim−1,fdim−1],

Note that imax_(k) and jmck_(k are repeated on each line of T) _(k) inthe file fich_centré. This makes it possible for these data,indispensable for the index change k→nk, to remain available after theaxis exchange operation.

(1604): The process is repeated as long as k is less than nbim.

(1401): Exchange of “plane” and “line” axes

The change of indices i→ni and j→nj having been carried out, it remainsto perform the change of index k→nk. To carry out this change of indexin a reasonable time, it is necessary to be able to load into centralmemory, rapidly, a plane (ni,k). For this operation to be possible, itis previously necessary to exchange the axes k and nj. This is what thisalgorithm does.

The file fich_centré previously created is reread, the order of the datais modified and a file fich_ech1 is written in which the data are in thefollowing order:

imax₀,jmax₀,T₀[0,0],T₀[1,0], . . . T₀[fdim−1,0],

imax₁,jmax₁,T₁[0,0],T₁[1,0], . . . T₁[fdim−1,0], . . .

imax_(nbim−1),jmax_(nbim−1),T_(nbim−1)[0,0],T_(nbim−1)[1,0], . . .T_(nbim−1)[fdim −1,0],

imax₀,jmax₀,T₀[0,1],T₀[1,1], . . . T₀[fdim−1,1],

imax₁,jmax₁,T₁[0,1],T₁[1,1], . . . T₁[fdim−1,1], . . .

imax_(nbim−1),jmax_(nbim−1),T_(nbim−1)[0,1],T_(nbim−1)[1,1], . . .T_(nbim−1)[fdim−1,1], . . .

imax₀,jmax₀,T₀[0,fdim−1],T₀[1,fdim−1], . . . T₀[fdim−1,fdim−1],

imax₁,jmax₁,T₁[0,fdim−1],T₁[1,fdim−1], . . . T₁[fdim−1,fdim−1], . . .

imax_(nbim−1),jmax_(nbim−1),T_(nbim−1)[0,fdim−1],T_(nbim−1)[1,fdim−1], .. . T_(nbim−1)[fdim−1,fdim−1,]

In other words, the axes nj and k are exchanged. This axis exchangeoperation is carried out block by block. FIG. 22 represents symbolically(1700) the content of a three-dimensional file corresponding to indicesi,j,k, arranged in memory horizontal plane by horizontal plane and eachplane being arranged line by line, one line being in the direction ofthe depth in the drawing. The axes i,j,k and the origin 0 of thecoordinate system are indicated in the drawing. The content of the fileobtained by inverting the axes j and k is represented in (1702). Thetransfer of data from one file to another takes place block by block,block (1701) being copied into central memory and then transferred at(1703). The reading and writing of the blocks takes place horizontalplane by horizontal plane, a horizontal plane in the file read notcorresponding to a horizontal plane in the file written. The size of theblock is the maximum size that the internal memory of the computer canhold. The operation is repeated for all the blocks, the blocks located“on the edge” generally not having the same dimension as the others.This procedure enables the exchange of axes with a computer whoseinternal memory size is smaller than that of the files used.

The procedure in C language (Microsoft C/C++ under Windows 95) whichpermits this operation is the following:

void echange_axes(FILE* read_file,FILE* write_file,int ktot,int jtot,intitot,int memory_limit)

{

int knum,jnum,bknum,bjnum,keff,jeff,k,j,bk,bj;

char* buff;

knum=(int)sqrt(((double)memory_limit)/((double)itot));

jnum=knum;

buff(char*)malloc(itot*knum*jnum);

bknum=ktot/knum;

if knum*bknum!=ktot) bknum+=1;

bjnum=jtot/jnum;

if(jnum*bjnum!jtot) bjnum+=1;

for (bk=0;bk<=bknum−1;bk++)

for (bj=0;bj<=bjnum−1;bj++)

{

if (bk==(bknum−1)) keff=ktot−knum*(bknum−1); else keff−knum;

if (bj==(binum−1)) jeff=tot−jnum*(bjnum−1); else jeff=num;

for (k=0;k<=keff−1;k++)

for (j=0;j<=jeff−1;j++)

{

fseek(read_file,(bk*knum+k)*jtot*itot+(bj*jnum+j)*itot,SEEK_SET);

fread(buff+k*jeff*itot+j*itot, 1, itot, read_file);

}

for (j=0;j<=jeff−1;j++)

for (k=0;k<=keff−1;k++)

{

fseek(write_file,(bj*jnum+j)*ktot*itot+(bk*knum+k)*itot,SEEK_SET);

fwrite(buff+k*jeff*itot+j*itot, 1, itot, write_file);

}

}

free(buff);

}

The parameters to be passed to this procedure are:

read_file: pointer to filed fich_centré

write_file: pointer to file fich_ech1

ktot: total number of images nbim

jtot: total number of lines in the frequency representation: fdim

itot: size in bytes of a line: fdim*sizeof(complex)+2*sizeof(int), wheresizeof(complex) designates the size in bytes of a complex number(T_(k)[ij]for example) and sizeof(int) the size in bytes of an integernumber (imax_(k) for example).

memory_limit: maximum size in bytes of the random-access memory (RAM)available to the procedure for the storage of blocks.

The files must be opened in “committed” mode, i.e. the read/writeoperations are carried out directly from or towards the hard disk,without intermediate buffering in central memory.

(1402): Calculation phase

The purpose of this calculation phase is to replace the “image” index kby the index nk given by the formula:${nk} = {\frac{fdim}{2} + \sqrt{K^{2} - \left( {i - C_{x}} \right)^{2} - \left( {j - C_{y}} \right)^{2}} - \sqrt{K^{2} - \left( {{imax}_{k} - C_{x}} \right)^{2} - \left( {{jmax}_{k} - C_{y}} \right)^{2}}}$

where i,j are the coordinates in the original system of coordinates(before centering). In the centered system of coordinates we thus have:${nk} = {\sqrt{\begin{matrix}{K^{2} - \left( {{ni} - {{fdim}\text{/}2} + {imax}_{k} - C_{x}} \right)^{2} -} \\\left( {{nj} - {{fdim}\text{/}2} + {jmax}_{k} - C_{y}} \right)^{2}\end{matrix}} - \sqrt{\begin{matrix}{K^{2} - \left( {{imax}_{k} - C_{x}} \right)^{2} -} \\\left( {{jmax}_{k} - C_{y}} \right)^{2}\end{matrix}} + \frac{fdim}{2}}$

When the same indices (i,j,k) are obtained several time by thereplacement of the image index, the value taken for the correspondingelement of the three-dimensional frequency representation is the averageof the values for which the indices (i,j,k) are obtained.

A point of the three-dimensional frequency representation, withcoordinates (ni,nj,nk), can only be obtained by this change ofcoordinates from a given plane (ni,k) corresponding to its index nj. Theplanes (ni,k) can consequently be processed independently of each other.When, in a plane (ni,k), the index k has been replaced by the index nk,it is possible to directly take the inverse two-dimensional Fouriertransform of this plane before going on to the next plane. This is whatis done by this part of the program, whose algorithm is represented indetail in FIG. 23. Its main stages are:

(1800): The following elements are read in the file fich_ech1 andtransferred to internal memory:

imax₀,jmax₀,T₀[0,nj],T₀[1,nj], . . . T₀[fdim−1,nj],

imax₀,jmax₀,T₁[0,nj],T₁[1,nj], . . . T₁[fdim−1,nj], . . .

imax_(nbim−1),jmax_(nbim−1),T_(nbim−1)[0,nj],T_(nbim−1)[1,nj], . . .T_(nbim −1)[fdim−1,nj]

(1801): The arrays D_(nj) and Poids, having dimensions fdim×fdim, areinitialized to 0.

(1802): The following condition is tested:${\left( {{ni} - {{fdim}\text{/}2} + {imax}_{k} - C_{x}} \right)^{2} + \left( {{nj} - {{fdim}\text{/}2} + {jmax}_{k} - C_{y}} \right)^{2}} < \left( {K\quad \frac{o}{n}} \right)^{2}$

where o is the aperture of the microscope, and n the index of theoptical oil and of the cover glass used, i.e. approximately:$\frac{o}{n} = {\frac{1.25}{1.51}.}$

If the condition is true, the point corresponds to a frequency vectorwhich is not beyond the aperture of the objective and is thus within theobservable area.

(1803): The value nk is calculated by the formula:${nk} = {\sqrt{\begin{matrix}{K^{2} - \left( {{ni} - {{fdim}\text{/}2} + {imax}_{k} - C_{x}} \right)^{2} -} \\\left( {{nj} - {{fdim}\text{/}2} + {jmax}_{k} - C_{y}} \right)^{2}\end{matrix}} - \sqrt{\begin{matrix}{K^{2} - \left( {{imax}_{k} - C_{x}} \right)^{2} -} \\\left( {{jmax}_{k} - C_{y}} \right)^{2}\end{matrix}} + {{fdim}\text{/}2}}$

(1804): The corresponding frequency value is added to the array D_(nj).The corresponding element of the array of weights, which will be used tocalculate the average value, is incremented.

D _(nj) [ni,nk]+=T _(k) [ni,nj]

Poids[ni,nk]+=1

(1805): When all the indices ni,nk have thus been covered, the programgoes through all the indices ni and nk while testing the conditionPoids[ni,nk]≠0 and, each time this condition is met, it performs:${D_{nj}\left\lbrack {{ni},{nk}} \right\rbrack} = \frac{D_{nj}\left\lbrack {{ni},{nk}} \right\rbrack}{{Poids}\left\lbrack {{ni},{nk}} \right\rbrack}$

(1806): The program performs the inverse two-dimensional Fouriertransform of the array D_(nj).

(1807): It stores the transformed array in the output file fich_calc inthe following order:

D_(nj)[0,0],D_(nj)[1,0], . . . D_(nj)[fdim−1,0],

D_(nj)[0,1],D_(nj)[1,1], . . . D_(nj)[fdim−1,1], . . .

D_(nj)[0,fdim−1],D_(nj)[1,fdim−1], . . . D_(nj)[fdim−1,fdim−1]

(1403): Second exchanges of axes

At this level, it remains to carry out an inverse Fourier transform witha single dimension along the axis nj. In order to be able to carry outthis transformation in a reasonable time, it is necessary to previouslyexchange the axes nj and nk. The program will then be able to load intocentral memory complete planes (ni,nj) in order to process them.

In the file fich_calc the data are arranged in the following order:

D₀[0,0],D₀[1,0], . . . D₀[fdim−1,0],

D₀[0,1],D₀[1,1], . . . D₀[fdim−1,1], . . .

D₀[0,fdim−1],D₀[1,fdim−1], . . . D₀[fdim−1,fdim−1],

D₁[0,0],D₁[1,0], . . . D₁[fdim−1,0],

D₁[0,1],D₁[1,1], . . . D₁[fdim−1,1], . . .

D₁[0,fdim−11],D₁[1,fdim−1], . . . D₁[fdim−1,fdim−1], . . .

D_(fdim−1)[0,0],D_(fdim−1)[1,0], . . . D_(fdim−1)[fdim−1,0],

D_(fdim−1)[0,1],D_(fdim−1)[1,1], . . . D_(fdim−1)[fdim−1,1], . . .

D_(fdim−1)[0,fdim−1],D_(fdim−1)[1,fdim−1], . . .D_(fdim−1)[fdim−1,fdim−1]

This file is reread, and a file fich_ech2 is generated, in which thedata are rewritten in the following order:

D₀[0,0],D₀[1,0], . . . D₀[fdim−1,0],

D₁[0,0],D₁[1,0], . . . D₁[fdim−1,0], . . .

D_(fdim−1)[0,0],D_(fdim−1)[1,0], . . . D_(fdim−1)[fdim−1,0],

D₀[0,1],D₀[1,1], . . . D₀[fdim−1,1],

D₁[0,0],D₁[1,1], . . . D₁[fdim−1,1], . . .

D_(fdim−1)[0,1],D_(fdim−1)[1,1], . . . D_(fdim−1)[fdim−1,1], . . .

D₀[0,fdim−1],D₀[1,fdim−1], . . . D₀[fdim−1,fdim−1],

D₁[0,fdim−1],D₁[1,fdim−1], . . . D₁[fdim−1,fdim−1], . . .

D_(fdim−1)[0,fdim−1],D_(fdim−1)[1,fdim−1], . . .D_(fdim−1)[fdim−1,fdim−1]

This exchange of axes nj and nk is carried out by blocks as in theprevious exchange of axes. The same procedure is used, the parameters tobe processed being:

read_file: pointer to file fich_calc

write_file: pointer to file fich_ech2

ktot: fdim

jtot: fdim

itot: Size in bytes of a line: fdim*sizeof(complex) wheresizeof(complex) designates the size in bytes of a complex number.

memory_limit: as previously, the size in bytes of the available memory.

(1404): Last Fourier transformation

This procedure consists in performing the inverse Fourier transformalong the axis nj. It is an iterative process on the index nk. Thealgorithm of this part of the program is represented in FIG. 24. Itsbasic stages are the following:

(1900): The program loads the following values into internal memory:

D₀[0,nk],D₀[1,nk], . . . D₀[fdim−1,nk],

D₀[0,nk],D₁[1,nk], . . . D₁[fdim−1,nk], . . .

D_(fdim−1)[0,nk],D_(fdim−1)[1,nk], . . . D_(fdim−1)[fdim−1,nk]

(1901): The program generates an array E_(nk,ni) with one dimension:

E_(nk,ni)[nj]=D_(nj)[ni,nk]

It performs the inverse Fourier transform of this array, generating thearray {tilde over (E)}_(nk,ni)

(1902): It stores the results in the file fich_rep, in the followingorder:

{tilde over (E)}_(nk,0)[0],{tilde over (E)}_(nk,1)[0], . . . {tilde over(E)}_(nk,fdim−1)[0],

{tilde over (E)}_(nk,0)[1],{tilde over (E)}_(nk,1)[1], . . . {tilde over(E)}_(nk,fdim−1)[1], . . .

{tilde over (E)}_(nk,0)[fdim−1],{tilde over (E)}_(nk,1)[fdim−1], . . .{tilde over (E)}_(nk,fdim)−[fdim−1]

The file thus generated then contains the three-dimensionalrepresentation of the object in the format U[ni,nj,nk] in which thecomplex element U[ni,nj,nk] is arranged in the file fich__rep at theaddress (nk*fdim*fdim+nj*fdim+ni) counted from the start of the file,the addressing taking place by “complex number” type elements.

5.22. Display

The array U having been generated, its content can be displayed.

The simplest display consists in extracting a section, one of theindices being fixed at a constant value. On this section, the real part,the imaginary part or the modulus can be displayed. For example, toextract a section in an imaginary part with nk constant, the programfirst generates the array V[ni,nj]=Re(U[ni,nj,nk]) in which nk has afixed value, and in which Re(x) designates the real part of x. It thendetermines the minimum and maximum values of the array$V,{{i.e.\quad V_{\min}} =},{\min\limits_{{ni},\quad {nj}}\left( {V\left\lbrack {{ni},{nj}} \right\rbrack} \right)},$

$V_{\min} = {\min\limits_{{ni},\quad {nj}}{\left( {V\left\lbrack {{ni},{nj}} \right\rbrack} \right).}}$

It then generates an array of pixels to be displayed on the screen(bitmap), the gray level of a given pixel with coordinates ni,nj being:${{val}\left\lbrack {{ni},{nj}} \right\rbrack} = {\frac{{V\left\lbrack {{ni},{nj}} \right\rbrack} - V_{\min}}{V_{\max} - V_{\min}}.}$

Another display method consists in extracting projections therefrom. Thedisplayed image depends to a great extent on the type of projectionused.

For example, for a projection of the real part along the axis nk, byintegration, the array V will be represented with${{V\left\lbrack {{ni},{nj}} \right\rbrack} = {\sum\limits_{k}{{Re}\left( {U\left\lbrack {{ni},{nj},{nk}} \right\rbrack} \right)}}},$

where Re(x) designates the real part of x, the sum over nk being takenbetween two “limit” planes depending on what is to be represented, andthe array V being represented as previously.

For a projection of the real part along the axis nk, by extraction ofthe maximum in absolute value, the array V will be represented with:${V\left\lbrack {{ni},{nj}} \right\rbrack} = {\max\limits_{nk}\left( {{{Re}\left( {U\left\lbrack {{ni},{nj},{nk}} \right\rbrack} \right)}} \right)}$

In these two cases, it is possible to represent the array V on thescreen according to the method already used for the representation of asection, i.e.${{val}\left\lbrack {{ni},{nj}} \right\rbrack} = \frac{{V\left\lbrack {{ni},{nj}} \right\rbrack} - V_{\min}}{V_{\max} - V_{\min}}$

in which V_(min) and V_(max) are respectively the minimum and maximumvalues of V.

Even though the example used was of sections with nk fixed andprojections along nk, these sections and projections can be carried outalong any direction, including oblique directions.

As a significant part of the frequency representation was not acquired,there are a certain number of resulting defects in the images obtainedunder spatial representation, which depend on the type of imagerepresented. In general, the defects are less significant for a topview. In the particular case of the projection by integration, theproduced image will remain of good quality up to a limit angle of theprojection direction with respect to the vertical: the projectiondirection must form with the vertical an angle sufficiently smaller than${Arc}\quad {{\sin \left( \frac{ouv}{n_{v}} \right)}.}$

In the case of the projection by extraction of the maximum, defects willbe present whatever the angle of observation. On the other hand,projection by extraction of the maximum is less sensitive to gaussiannoise than projection by integration.

In the case of the projection by integration, a fast method may beobtained for the projection. This method requires storage in memory ofthe frequency representation of the object. The latter may have beenobtained, for example, as indicated in 5.21 but without taking theFourier transforms described in (1806) and in (1901). The fast methodcomprises two stages:

Stage 1: extraction, frequency representation, of a plane passingthrough the origin and orthogonal to the projection direction.

Stage 2: inverse Fourier transformation of this plane.

The two-dimensional array thus obtained constitutes a projection alongthe direction used to extract the frequency plane.

It is possible to generate a stereoscopic view by extracting twoprojections along appropriate directions and using the anaglyph method,for example, to view them. In this case, the use of the fast methodmakes it possible, if the computer has a sufficient memory for storingthe three-dimensional representation, to modify in “real time” theobservation angle of an image precalculated in frequency representationform.

6. DESCRIPTION OF A SECOND EMBODIMENT

This embodiment is a simple variant of the first and is represented inFIG. 25.

6.1. Principle

In the first embodiment, the sensor (118) is in the back (image) focalplane of the optical assembly made up of the objective (113) and lenses(115) and (117). The plane illuminating wave thus has a point image inthis plane, and a spherical reference wave centered virtually on theobject must be used to obtain homogeneous illumination on the sensor(118). In this second embodiment, the sensor (2018) is placed directlyin the image plane of the objective. A plane illuminating waveconsequently no longer has a point image. The reference wave must be theimage through the objective of a virtual plane wave passing through theobject.

An image in complex numbers is obtained on the CCD (2018) from threeimages differing in the phase of the reference wave, using, as in thefirst embodiment, the formula${{s\quad \frac{\overset{\_}{r}}{r}} = {{\frac{1}{6{r}}\left( {{2{s_{0}}^{2}} - {s_{1}}^{2} - {s_{2}}^{2}} \right)} + {j\quad \frac{1}{2\sqrt{3}{r}}\quad \left( {{s_{1}}^{2} - {s_{2}}^{2}} \right)}}}\quad$

The two-dimensional Fourier transform of this image gives an image incomplex numbers equivalent to that which, in the first embodiment, wasobtained directly in the plane of the CCD sensor. This image thusreplaces the image obtained directly on the sensor in the firstembodiment. For the rest, this embodiment uses the same principles asthe first.

6.2. Equipment description

The system is represented in FIG. 25. The elements of this figure,identical to those of FIG. 1, are numbered by replacing the first digit1 of the elements of FIG. 1 by the digit 20. For example (116) becomes(2016). This system is similar to the one used in the first embodiment,except that:

The device for controlled attenuation of the beam, made up of theelements (104) and (105) is eliminated.

The CCD (2018) is placed in the plane in which the diaphragm (114) waspreviously located, and consequently the elements (114) (117) (115) areeliminated.

The virtual image, after reflection on the semi-transparent mirror(2016), of the focussing point of the beam coming from the lens (2023),must be located on the optical axis and in the rear focal plane of theobjective (2013). The elements (2023)(2022)(2021)(2016) are thus movedso as to meet this condition.

On the CCD sensor CCD (2018), the size of the basic cell (distancebetween the central points of two neighboring pixels) must be less than${\frac{\lambda}{2}\quad \frac{g}{ouv}},$

where ouv designates the numerical aperture of the objective, λ thewavelength in vacuum of the laser used, g the magnification. Forexample, for an ×100 objective with an aperture of 1.25 we find 25micrometers. It is possible to use an objective modified to have an ×50magnification, in order to obtain 12 micrometers, thus allowing the useof a current camera with 10-micrometer graduations in a more optimummanner than with an ×100 objective.

The lens (2023) is mounted on a positioner allowing translation alongthe axis of the beam entering this lens.

6.3. Adjustments: overall view

In the preceding embodiment, the image received on the sensor was in thefrequency domain. An image in the spatial domain could if necessary beobtained from that frequency domain image by two-dimensional inverseFourier transform, which was done by the focussing program described inFIG. 14. In this second embodiment, the image received on the sensor isin the spatial domain and an image in the frequency domain can beobtained by Fourier transform.

Each time the two-dimensional frequency image received directly on theCCD sensor (118) was used, it is necessary to use the two-dimensionalFourier transform of the image received on the CCD sensor (2018), whichconstitutes a frequency image which can have a dimension of 256×256square pixels. Conversely, the focussing program must be replaced by aprogram for direct display of the image received on the CCD sensor.

As the beam attenuation device is eliminated, a single image must beused instead of two in the stages in which this device was used.

The camera (2019) is fixed. The adjustment of this camera's positionalong the optical axis is replaced by an adjustment of the position ofthe lens (2023) along the axis of the beam entering this lens. Thecamera position adjustment in the plane orthogonal to the optical axisis replaced by an angular adjustment of the mirror (2016).

For the rest, the operating procedure is similar to that of the previoussystem. The adjustment stages and the programs used are given in detailbelow:

6.4. Current usage program

In addition to the programs described in 5.5., use is made of afrequency image display program. This program generally replaces thedirect display program used in the first embodiment, which allowed theobservation of an image in the frequency domain. To use this program, areference wave must be used, whereas in the direct display program usedin the first embodiment it was not necessary. When this program is used,it is necessary to avoid vibrations and hence to release at leasttemporarily the focussing device if it is used, or wait for theabsorption of the vibration before each image when stepping motors areused.

This program is similar to the focussing program described in 5.14 andwhose algorithm is given in FIG. 14. It is modified as follows:

The image acquisition stage (1000), represented in detail in FIG. 13, ismodified as indicated in FIG. 26, in order to take into account theabsence of the beam attenuation device.

The stage (1001) is eliminated.

The stage (1002) is modified to take into account the absence of thearray H. For each pixel, the value is generated according to thefollowing equations:${S\left\lbrack {i,j} \right\rbrack} = \left\lbrack \quad {{\frac{1}{6}\left( {{2{I\left\lbrack {0,i,j} \right\rbrack}} - {I\left\lbrack {1,i,j} \right\rbrack} - {I\left\lbrack {2,i,j} \right\rbrack}} \right)} + {\overset{\sim}{j}\frac{1}{2\sqrt{3}}\left( {{I\left\lbrack {1,i,j} \right\rbrack} - {I\left\lbrack {2,i,j} \right\rbrack}} \right)}} \right\rbrack$

During the stage (1002) the program limits the array S to dimensions ofhpix×hpix but does not perform any averaging.

The stage (1004) is replaced by a direct Fourier transform.

During the stage (1005), the program displays the intensity,corresponding to the square of the modulus of the elements of thetransformed array S, as well as the maximum value of this intensity, thecoordinates of the corresponding point and the ratio between theintensity of this point and the sum of the intensities of its eightneighbors.

This program makes it possible to determine the point nature(punctuality) and the general appearance of a frequency image. On theother hand, to determine non-saturation (which must be checked in almostall the stages, a fact which will no longer be mentioned), the programsdescribed in 5.5. continue to be used, and the non-saturation must beverified directly on the CCD sensor CCD (2018).

6.5. Adjustment of position of laser (2000) and mirror (2021)

This stage is similar to the stage described in 5.6.

6.6. Adjustment of piezoelectric actuator control voltages

The process is the same as the one described in 5.12., except that thereis no adjustment of the diaphragm (114) which is eliminated and theposition of (2009) is adjusted so as to maximize the intensity receivedon the sensor. The fact that the image received directly by the sensoris in the spatial domain does not affect the result.

6.7. Adjustment of reference wave level

This stage is identical to the one described in 5.11, the level of thereference wave being measured on the direct image.

6.8. Position adjustment of (2023) (2002) (2016)

This adjustment is similar to the one described in 5.7. The directdisplay program is replaced by the frequency image display programdescribed in 6.4, for which the presence of the reference wave isnecessary. The adjustment of the camera position along the optical axisis replaced by an adjustment of the position of the lens (2023) alongthe axis of the beam entering this lens. The camera position adjustmentin the plane orthogonal to the optical axis is replaced by an angularadjustment of the mirror (2016).

6.9. Adjustment of condenser position (2011)

This adjustment is similar to the one described in 5.8. but it is thefrequency image which must be observed in the presence of the referencewave, and not the direct image in the absence of the reference wave.

6.10. Adjustment of position of lens (2006)

This adjustment is similar to the one described in 5.9. but it is thefrequency image which allows punctuality to be evaluated.

6.11. Determination of number of steps per pixel

This stage is similar to the one described in 5.10. but the maximumintensity pixel is observed on the frequency image.

6.12. Obtaining the constant K

This stage is carried out on the same principle as the one described in5.14. but is modified to take into account the inversion between directimage and frequency image.

To obtain the image of the micrometer, the objective is first placed ina roughly focussed position, the micrometer having been introduced asthe object. The program for the direct display of the image received onthe CCD is started, in the absence of a reference wave. The diaphragm at(2007) must be adjusted so that the observed area appears uniformlyilluminated. The micrometer is moved under the objective until an imagethereof is obtained.

The microscope objective is then moved by the focussing device so as toobtain a correctly focussed image. To facilitate focussing, it isadvantageous to display a part of the micrometer in which lines ofdifferent lengths are present. This limits “false focussing” due tointerference phenomena ahead of the micrometer.

When a good image has been obtained, the program is stopped and theobtained image is used to obtain the distance in number of pixelsbetween two lines, as in 5.14. If the distance between two graduationsseparated by D_(real) micrometers is on the image thus obtained ofD_(pix) pixels, if the nominal index of the objectives is n_(v) (ingeneral, n_(v) is close to 1.5) and if the wavelength of the laser invacuum is λ (λ=0.633 micrometers) and if the number of points of theFourier transform which will be used for image acquisition is N (N=256)then we have:${K = {\frac{n_{v}}{\lambda}\frac{N}{D_{pix}}D_{real}}},$

in which D_(real) and λ are in the same unit.

6.13. Adjustment of diaphragm

This stage is similar to the one described in 5.15. but the focussingprogram is replaced by a direct display of the image received on thesensor in the absence of a reference wave.

6.14. Recording of reference wave

This stage is similar to the one described in 5.16. The reference wavealone is recorded on the direct image.

6.15. Focussing on studied object

This stage is simplified, the specific focussing program being replacedby a program for the direct display of the image received on the CCDsensor in the absence of a reference wave.

6.16. Adjustment of condenser position and adjustment of diaphragms

This stage is similar to the one described in 5.18. but:

the frequency image is now obtained in the presence of the referencewave by Fourier transformation of the image received directly on the CCDsensor, and not directly as in 5.18.;

by contrast, the image of the object in the spatial domain is nowobtained directly and not by Fourier transformation of the imagereceived on the CCD sensor.

6.17. Adjustment of filters

The filters are adjusted so that the image formed on the CCD by theilluminating wave alone has a maximum intensity of 64.

6.18. Image acquisition

This stage is similar to the one described in 5.20. but the followingmodifications must be taken into account:

stage (1302) is replaced by the acquisition of images described in FIG.26

stage (1303) is eliminated.

stage (1304) is modified: the complex numbers representation S[i,j] isgenerated by assigning to each point the following value:$\left. {{S\left\lbrack {i,j} \right\rbrack} = {\left\lbrack {{\frac{1}{6\sqrt{{Iref}\left\lbrack {i,j} \right\rbrack}}\left( {{2\quad {I\left\lbrack {0,i,j} \right\rbrack}} - {I\left\lbrack {1,i,j} \right\rbrack}} \right)} - {I\left\lbrack {2,i,j} \right\rbrack}} \right) + {\overset{\sim}{j}\quad \frac{1}{2\sqrt{3\quad {{Iref}\left\lbrack {i,j} \right\rbrack}}}\left( {{I\left\lbrack {1,i,j} \right\rbrack} - {I\left\lbrack {2,i,j} \right\rbrack}} \right)}}} \right\rbrack$

then by taking the Fourier transform of array S.

6.19. Calculation stage

This stage is identical to the one described in 5.21.

6.20. Display

This stage is identical to the one described in 5.22

7. DESCRIPTION OF A THIRD EMBODIMENT

This embodiment is more complex and more costly than the preceding butallows higher performance in terms of definition and speed.

7.1. Principle

This acquisition method makes it possible to improve the performances ofthe first embodiment as follows:

Higher image acquisition speed:

In the first embodiment, this speed is limited by the mechanicalmovement of the stepping motors and the need to wait for the absorptionof vibrations induced after each movement. The present embodiment allowsthe acceleration of this image acquisition by replacing this mechanicalsystem by an optical beam deflection system based on liquid crystals andnot inducing any mechanical displacements in the system.

Improved precision:

In the first embodiment, the precision is limited by the impossibilityof adopting all the possible directions for the illuminating beam and bythe fact that the reflected wave is not taken into account. The presentembodiment uses a two-objective system. Illumination then takes placethrough an objective, thus enabling the frequency vector of theilluminating wave to vary on both sphere portions limited by theaperture of each objective. In addition, the reflected wave passesthrough the illuminating objective again and can be taken into account.

In the first embodiment, the variations in the intensity of thediffracted wave as a function of the polarization direction of theilluminating wave are not taken into account, thus resulting in errorsin the measurement of high frequencies. In the present embodiment,polarization rotators allow the variation of the illuminating wavepolarization direction and the diffracted wave analysis direction. Analgorithm takes into account all the measurements thus obtained in orderto obtain frequency representations in which dependence of thediffracted wave in relation to the polarization of the illuminating wavehas been removed.

Compensation for spherical aberration:

In the preceding embodiments, the average index in the observed samplemust be close to the nominal index of the objective. Otherwise, thedifference between the average index of the sample and the nominal indexof the objective results in a spherical aberration which significantlylimits the thickness of the observable sample. In this new embodiment,the equipment configuration and the algorithms used permit thecompensation of phase differences induced by the average index of thesample and the cancellation of this spherical aberration.

Paragraph 7.2. describes materially the microscope used.

This microscope undergoes a number of preliminary adjustments carriedout in the absence of the observed sample and which normally do not haveto be repeated when the observed sample is modified:

The position of the different system elements is adjusted as describedin Paragraph 7.4.

The modulus of the reference wave is determined as described inParagraph 7.4.

The parameters K_(p), equivalent to the parameter K used in the firstembodiment, are determined as described in Paragraph 7.6.

The characteristics of the illuminating beams used are determined asdescribed in Paragraph 7.9.

The arrays of control indices are determined as described in Paragraph7.13.

After introducing the sample, the microscope undergoes a second seriesof adjustments:

The position of the objectives is adjusted as described in Paragraph7.10.

The relative coordinates x,y,z of the points of origin of the referencebeams associated with each objective, as well as the average index n₀ ofthe sample and its thickness L, are determined as described in Paragraph7.11. This determination implies the use of a specific algorithmdescribed in Paragraph 7.8., using equations established in Paragraph7.7. A simplified version of this algorithm is also used in Paragraph7.9.

The value w₀ characterizing the position of the sample is calculated asdescribed in Paragraph 7.15.

The determination of this value makes use of an image acquisitionprocedure described in Paragraph 7.12. and of equations established inParagraph 7.14. Simultaneously, a first adjustment of the position ofthe sample is carried out as indicated in Paragraph 7.15.3.

The aberration compensation function, in the form of arrays D_(p), isobtained as described in Paragraph 7.16.

When these preliminary adjustments have been carried out, the procedurefor obtaining three-dimensional representations can be started. Thisprocedure is described in 7.17. It makes use of the image acquisitionprocedure described in 7.12. and uses the array D_(p) determined in7.16. By repeating this procedure indefinitely, it is possible to obtaina succession of three-dimensional representations characterizing thetime evolution of the observed sample. The position of the sample mustbe adjusted so that the obtained representation is that of a region ofinterest of the sample. This adjustment is carried out as indicated in7.17.3. and may involve a repetition of the preliminary calculationstages for w₀. and D_(p) described respectively in 7.15.2. and in 7.16.

Different variants of the algorithms and adjustments carried out aredescribed in 7.18. Many adjustments can be eliminated if conditions arefavorable to this, for example if the index and the thickness of thesample are known in advance.

A design method for microscope objectives, specifically suited to thismicroscope, is described in Paragraph 7.19.

7.2. Equipment description

7.2.1. Overall view

FIGS. 27 and 28 constitute an overall view of the system. The greaterpart of the system, represented in FIG. 27, is within a horizontal planeand is supported by an optical table. However, the two microscopeobjectives used must be positioned on a vertical axis (2263) to be ableto use a sample (2218) positioned horizontally. The axis (2263) is atthe intersection of two vertical planes defined moreover by theirhorizontal axes (2261) and (2262). These horizontal axes can form anangle of 0 degrees, 90 degrees or 180 degrees between them. FIG. 28represents partially a section along the vertical plane defined by(2261) and partially a section along the vertical plane defined by(2262).

The beam coming from a laser (2200) polarized in the vertical directionwill be divided into four beams supplying the right and left opticalchains associated with the two objectives of the microscope. These fourbeams are designated in the diagram and in the text by the followingabbreviations:

FRD: right-hand reference beam.

FRG: left-hand reference beam.

FED: right-hand illuminating beam.

FEG: left-hand illuminating beam.

Each of these beams will be divided subsequently into a main beam, whichwill be designated the beam of origin, and an opposite indicator beam.The opposite indicator beams will be denoted FRDI, FRGI, FEDI, FEGI.

The beam comes from the laser (2200) and has its electric field vectordirected along an axis orthogonal to the plane of the figure. It passesthrough a beam expander (2201) and is then separated into anilluminating beam and a reference beam by a semi-transparent mirror(2202). The illuminating beam passes through a diaphragm (2248), afilter (2203) and then a beam attenuation device (2204), a phase shiftdevice (2205), a beam deflection device (2206) which allows thedirection of this parallel beam to be varied. It is then deflected by asemi-transparent mirror (2207) which separates a right-hand illuminatingbeam and a left-hand illuminating beam, intended to illuminate thesample in two opposite directions. The right-hand illuminating beam FEDis deflected by a mirror (2208), passes through a beam deflection andswitching device (2209), a phase rotator (2210), and is separated by asemi-transparent mirror (2211) into a main illuminating beam and anopposite indicator beam. The main illuminating beam then passes throughan achromat (2212), a diaphragm (2213), is reflected on a mirror (2214)sending it upwards, and then on mirrors (2215) and (2216), and passesthrough the objective (2217) to illuminate the sample (2218). Afterpassing through the sample, it passes through the objective (2219), isreflected by the mirrors (2220) (2221) (2222), and then passes throughthe diaphragm (2223), the achromat (2224), the semi-transparent mirror(2225), the phase rotator (2226), the achromat (2227), thesemi-transparent mirror (2228) and the polarizer (2253) and is receivedby the CCD sensor (2229).

The two objectives (2217) and (2219) must have their optical axis (2263)vertical so that the optical oil required for their use does not flow.The mirrors (2214) (2215) (2216) (2220) (2221) (2222) have the utilityof deflecting the beam so that it can pass through the objectives in avertical direction. FIG. 28 represents a section along the axes (2262)and (2263), articulated around the optical axis (2263).

FIG. 29 is a representation of the optical path of the light raysbetween the objective (2219) designated by “OM” and the CCD sensor(2229) designated by “CCD”. The mirrors, semi-transparent mirrors andphase rotator have been omitted in the figure but influence the positionof the different elements. The rays are propagated along an optical axisshown as “straight”, which in reality ceases to be straight between theplanes P2 and P1, a zone in which it deflected by mirrors (2220) (2221)(2222) to approach the objective in a vertical plane. The left-hand partof the figure represents the optical path of rays which are parallel inthe studied sample, and the right-hand part opposite represents theoptical path of rays coming from a point in the observed area. Theachromat (2224) is designated by “L1”, the diaphragm (2223) by “D”, theachromat (2227) by “L2”. f₁ is the focal length of L1, f₂ is the focallength of L2. P1 is the plane in which are focussed rays enteringparallel into the objective (rear focal plane). This plane must coincidewith the front focal plane of the achromat (L1) so that a ray parallelon entering the objective is also parallel between the achromats L1 andL2. P2 is the plane in which the image of the observed sample is formed.It is in this plane that the diaphragm (D) must be positioned. P3 is thevirtual image of the plane P2 through the achromat L1. P3 must coincidewith the front focal plane of L2 so that a ray coming from a centralpoint of the observed object, forming a point image in P2, reaches theCCD in the form of a parallel ray. P6 is the rear focal plane of L2. Itis in this plane that the CCD must be placed so that a ray parallel onentering the objective forms a point image on the CCD.

The optical path of rays between the objective (2217) and the sensor(2239) is symmetrical with the preceding.

The reference beam separated by the mirror (2202) is reflected by(2233), passes through a filter (2234), and a semi-transparent mirror(2235) transforming it into a left-hand part and a right-hand part. Theleft-hand part is reflected by mirrors (2254) (2236), passes through thecomplementary filter (2255), the phase shift device (2251) and thediaphragm (2250) and then reaches the semi-transparent mirror (2228)which separates it into a main beam and an opposite indicator beam. Themain beam is directed towards the CCD (2229).

Both the illuminating beam and the reference beam have oppositeindicator beams having the same characteristics as the main beam butdirected in the opposite direction. The opposite indicator FRGI of thereference beam FRG, coming from the semi-transparent mirror (2228),passes through the achromat (2281) and is focussed on a mirror (2282)which reflects it. It then passes against through the achromat (2281)which again makes it parallel, then it is again reflected by thesemi-transparent mirror (2228). It then has the same (but directedopposite) direction as the reference beam directed towards the CCDsensor (2229). Similarly, the opposite indicator FRDI of theilluminating beam FRD, coming from the semi-transparent mirror (2211),is focussed by the achromat (2231) on the mirror (2232). The latterreflects it, and after another reflection on the semi-transparent mirror(2211), it has the direction opposite that of the main illuminating beamdirected towards the objective (2217).

The entire device is optically symmetrical in relation to the observedobject. There is thus a left-hand illuminating beam having a symmetricalrole in relation to the right-hand illuminating beam, and a right-handreference beam having a symmetrical role in relation to the left-handreference beam.

The left-hand illuminating beam FEG, coming from the semi-transparentmirror (2207), is reflected on the mirrors (2280) and (2283) and thenpasses through the deflection and switching device (2240) equivalent to(2209). It then passes through the polarization rotator (2241), isseparated by the semi-transparent mirror (2225) into a main beam whichis directed towards the microscope objective (2219), and an oppositeindicator beam FEGI which passes through the achromat (2242), isfocussed on the mirror (2243) and finally reflected again on (2225).

The right-hand reference beam FRD, coming from the semi-transparentmirror (2235), is reflected by the mirror (2244) and passes through thecomplementary filter (2256). The semi-transparent mirror (2245)separates it into a main beam which passes through the polarizer (2252)and reaches the CCD (2239), and an opposite indicator beam FRDI whichpasses through the achromat (2246) and is focussed on the mirror (2247),and then returns towards the semi-transparent mirror (2245) whichreflects it in the direction of the objective (2217).

The polarizers (2252) and (2253) are plates of small thicknessconsisting of a dichroic sheet maintained between two glass plates.

The zones (2274) (2275) (2276) (2277), delimited by dotted lines in thedrawing, correspond to parts of the system immersed entirely in theoptical oil. Such a zone consequently constitutes a leaktight recipientcontaining the optical elements visible in the drawing. The entry andexit of the beam into this recipient take place through windows havingreceived an antireflecting treatment on their outer face. This makes itpossible to limit defects related to the cutting of the glasses used inthe different devices included therein.

The CCDs (2239) and (2229) are integrated in cameras (2284) and (2230)themselves fixed to three-axis positioners allowing their adjustmentalong the axis (2264) and along the two axes orthogonal to (2264), aswell as in rotation around the axis (2264). The achromats (2227) (2224)(2212) (2237) (2246) (2231) (2242) (2281) are fixed to positioners withone axis allowing fine adjustment of the position in the direction ofthe axis (2264). The mirrors (2282) (2243) (2232) (2247) are fixed topositioners allowing an adjustment of their orientation. The diaphragms(2213) and (2223) are adjustable and fixed to two-axis positionersallowing the adjustment of their position in the plane orthogonal to(2264). The semi-transparent mirrors (2225) (2228) (2211) (2245) arefixed to positioners allowing the adjustment of their orientation. Themirrors (2214) and (2222) are fixed to positioners allowing theadjustment of their orientation. The microscope objective (2219) isfixed to a two-axis positioner allowing it to be moved in a planeorthogonal to the axis (2263). The objective (2217) is fixed to afocussing device allowing it to be moved along the axis (2263). Theentire system is fabricated with the greatest precision possible in thepositioning of the various elements.

The mirrors (2247) (2232) (2243) (2282) are equipped with manualshutters (2257) (2258) (2259) (2260) allowing the suppression of thebeams reflected by these mirrors. The beams FRD and FRG can besuppressed by using completely opaque filters.

The sample (2218) is made up of two cover glasses of standard thickness(150 μm) between which there is a thin layer (50 to 100 μm) of thesubstance to be observed. This sample is fixed to a thicker plate, insuch manner that the latter does not prevent the access of the samplethrough the objectives. The assembly is fixed to a three-axistranslation positioner.

The objectives used can be, for example, planapochromatic objectiveswith a numerical aperture ouv=1.4 and magnification g=100 forming theimage at 160 mm from the shoulder of the objective. It is also possibleto use other types of objectives, described in Paragraphs 7.19 to 7.21.

The achromats (2212) (2237) (2224) (2227) (2246) (2231) (2242) (2281)can have, for example, the same focal length f=f₁=f₂=200 mm.

The CCD sensors used must have square pixels and a square useful regionhaving a side length of ${D = {2\quad f_{2}\quad \frac{ouv}{g}}},$

the number of pixels being N_(pix)×N_(pix) with, for example,N_(pix)=256. The beam control devices (2204) (2205) (2206) (2209) (2240)(2251) (2210) (2241) (2226) (2238) are all driven by phase rotatorscontrolled by bipolar voltages. The control of these devices must besynchronized with the acquisition of the images by the camera. Thecamera can be a fast camera of the movement analyzer type, provided withsufficient memory, available for example from Kodak. The calculationsystem is a computer equipped with a sufficient memory to store therequired three-dimensional arrays. Machines having 8 GB of memory forexample are available from Digital Equipment.

The filters at (2203) (2234) (2255) (2256) allow the adjustment of theintensity of the different beams. As in the first embodiment, theirvalues must be frequently adjusted during the different settingoperations and during the use of the microscope. These adjustments arecarried out in a similar fashion to what was done in the firstembodiment and will not be mentioned again. They are also designed tolimit the intensity of the beams which move in a direction opposite tothe normal and tend to return towards the laser (2200), during certainadjustment operations.

7.2.2. Beam attenuation device

The attenuation device is represented in FIG. 30. It consists of a phaserotator (2501) designated “R1” in the figure, a Glan-Thomson polarizer(2502) designated “POL1”, a second rotator (2503) designated “R2”, and asecond polarizer (2504) designated “POL2”. The beam entering the deviceis vertically polarized. The angle of the neutral axis of (2501) withthe vertical is θ for a bipolar voltage of −5 V applied to the terminalsof the device and turned by an angle α by applying a voltage of +5 V,with α=22 degrees approximately. The neutral axis of the rotator (2503)is characterized by the same angles, but in relation to the horizontaldirection and not to the vertical direction. The polarizer (2502)selects the horizontal polarization direction. The polarizer (2504)selects the vertical polarization direction.

FIG. 31 illustrates the functioning of the part of the device consistingof (2501) and (2502) for an applied voltage of −5 V. It represents, inbold lines, the electric field vector (2505) of the beam entering thedevice, in a coordinate system consisting of the vertical polarizationaxis (2506) and the horizontal polarization axis (2507). Passage throughthe rotator R1 (2501) causes this vector to turn by an angle of 2θ andit is thus transformed into (2508). Passage through the polarizer POL1(2502) constitutes a projection on to the horizontal axis. At the exitof this polarizer, the electric field vector (2509) of the beam is thushorizontal and its amplitude has been multiplied by a factor sin(2θ).

FIG. 32 illustrates the functioning of the part of the device consistingof (2501) and (2502) for an applied voltage of +5 V. The neutral axis ofR1 having been turned by an angle α, the electric field vector is turnedby a total angle of 2(α+θ) and the electric field amplitude at the exitis multiplied by sin(2θ+2α). The attenuation factor between the “open”position (+5 V) and the “closed” position (−5 V) is thus$a_{1} = {\frac{\sin \quad \left( {2\quad \theta} \right)}{\sin \quad \left( {{2\alpha} + {2\quad \theta}} \right)}.}$

This expression is inverted into:$\theta = {\frac{1}{2}\quad {Arc}\quad \tan \quad \left( \frac{\sin \quad \left( {2\quad \alpha} \right)}{\frac{1}{a_{1}} - {\cos \quad \left( {2\quad \alpha} \right)}} \right)}$

For example for $a_{1} = \frac{1}{16}$

and α=22° we find θ=1.30°

The second part of the device, made up of (2503) and (2504), functionsexactly like the first, except that it takes a horizontally polarizedbeam at the entrance and delivers a vertically polarized beam at theexit. The attenuation factor a₂ of this second part is thus given by thesame formula as a₁ and the two parts of the device will be adjusted soas to have the same (or roughly the same) attenuation in each part.Owing to adjustment disparities between the two parts of the device, a₂and a₁ are however not rigorously equal in practice.

The attenuation control takes place in accordance with the table below,where V₁ designates the bipolar voltage applied to (2501) and V₂ thatapplied to (2503).

V1 V2 attenuation −5 V −5 V α₁ α₂ −5 V +5 V α₁ +5 V −5 V α₂ +5 V +5 V 1

7.2.3. Phase shift device

The phase shift device is made up of two identical units placed oneafter the other. One unit is made up as indicated in FIG. 33.

The vertically polarized beam at the entrance of the device first passesthrough a phase rotator (2601) designated “R1”, then through a uniaxialbirefringent plate (2602) designated “LP”, then a second phase rotator(2603) designated “R2” and a polarizer (2604) designated “POL”. The twopositions of the neutral axis of each rotator are arranged symmetricallyin relation to the vertical axis. The positions of the two rotatorscorresponding to a given control voltage are on the same side of thevertical axis: for a voltage of −5 V they are represented in dottedlines, for a voltage of +5 V they are represented by solid lines.Similarly, the two axes of the birefringent plate are arrangedsymmetrically in relation to this vertical axis (the third axis being inthe beam propagation direction). FIG. 34 shows the condition of theelectric field vector of the beam at each stage of passage through thedevice for a voltage of −5 V applied to each rotator.

FIG. 35 is based on FIG. 34 but giving the values of the angles betweenthe different vectors and the phase differences between these vectorsand the vector at the entrance of the device. FIG. 36 is based on FIG.34 but gives the values of the angles between the different vectors andthe attenuation on each vector.

The electric field vector (2605) at the entrance of the device isvertical. After passing through the rotator R1 (2601) it is symmetrizedin relation to the neutral axis (2606) of this rotator, giving thevector (2607). After passing through the birefringent plate, the vector(2607), represented in dotted lines, is broken down into two components(2608) and (2609) corresponding to each neutral axis of the plate. Thecomponent (2609) is affected by a phase shift φ/2 and is reduced inamplitude by a factor$\cos \quad {\left( {\frac{\pi}{4} - \alpha} \right).}$

The component (2608) is affected by a phase shift φ/2 and is reduced inamplitude by a factor$\sin \quad {\left( {\frac{\pi}{4} - \alpha} \right).}$

After passing through the rotator (2603) the assembly is symmetrized inrelation to the neutral axis (2612) of this rotator. (2608) istransformed into (2611), while (2609) is transformed into (2610). Afterpassing through the polarizer (2604), these two components are projectedon to a vertical axis. The component (2610) is multiplied by a factor$\cos \quad \left( {\frac{\pi}{4} - \alpha} \right)$

and has thus been affected globally by a factor$\cos^{2}\quad {\left( {\frac{\pi}{4} - \alpha} \right).}$

The component (2611) is multiplied by a factor$\sin \quad \left( {\frac{\pi}{4} - \alpha} \right)$

and has thus been affected globally by a factor$\sin^{2}\quad {\left( {\frac{\pi}{4} - \alpha} \right).}$

The two are then added to give a single component (2615) with the value:$v = {{\cos \quad \left( {{\omega \quad t} - \frac{\phi}{2}} \right)\quad \cos^{2}\quad \left( {\alpha - \frac{\pi}{4}} \right)} + {\cos \quad \left( {{\omega \quad t} + \frac{\phi}{2}} \right)\quad \sin^{2}\quad \left( {\alpha - \frac{\pi}{4}} \right)}}$

where ω is the pulse of the wave, t is the time. We verify:$v = {{\cos \quad ({wt})\quad \cos \quad \frac{\phi}{2}} + {\sin \quad ({wt})\quad \sin \quad \frac{\phi}{2}\quad \sin \quad \left( {2\quad \alpha} \right)}}$

or in complex representation:$c = {{\cos \quad \frac{\phi}{2}} + {j\quad \sin \quad \frac{\phi}{2}\quad \sin \quad \left( {2\quad \alpha} \right)}}$

If θ is the argument of c, we then have:${\cos \quad \theta} = \frac{\cos \quad \frac{\phi}{2}}{\sqrt{{\sin^{2}\quad \left( {2\quad \alpha} \right)} + {\cos^{2}\quad \frac{\phi}{2}\quad \left( {1 - {\sin^{2}\quad \left( {2\quad \alpha} \right)}} \right)}}}$${\sin \quad \theta} = \frac{\sin \quad \frac{\phi}{2}\quad \sin \quad \left( {2\quad \alpha} \right)}{\sqrt{1 + {\sin^{2}\quad \frac{\phi}{2}\quad \left( {{- 1} + {\sin^{2}\quad \left( {2\quad \alpha} \right)}} \right)}}}$

These formulas are inverted as:${\sin \quad \frac{\phi}{2}} = \frac{\sin \quad \theta}{\sqrt{{{\sin^{2}\quad \theta} + {\cos^{2}\quad \theta \quad \sin^{2}\quad \left( {2\quad \alpha} \right)}}\quad}}$${\cos \quad \frac{\phi}{2}} = \frac{\cos \quad \theta \quad \sin \quad \left( {2\quad \alpha} \right)}{\sqrt{{\sin^{2}\quad \theta} + {\cos^{2}\quad \theta \quad \sin^{2}\quad \left( {2\quad \alpha} \right)}}}$

The creation of a phase shift θ=60° is sought in a position of therotators and θ=−60° in the symmetrical position, which corresponds to atotal phase shift of 120 degrees. The above equation makes it possibleto determine the value of the total phase shift ω created by the platebetween its two neutral axes.

With θ=60 and α=40 degrees we obtain: ω=120.7566 degrees. It is thusnecessary to use a uniaxial plate creating, for the consideredwavelength, a phase difference of 120.75 degrees between its two axes.

The two axes of the plate do not have a symmetrical role. If Δω_(i) isthe phase shift upon passing through the plate for a ray polarized alongthe axis i, there is only one choice for the axes 1 and 2 such thatΔω₁−Δω₂≡120 degrees. The uniaxial plate must be oriented so that therotator (2601) causes the turning of the polarization of the incidentray towards the axis 2 when it is subjected to a bipolar voltage of −5V.

With the phase shift device made up of two units of this type, letV_(ij) be the voltage applied to the i-th rotator of the j-th device (iand j varying from 1 to 2). The control of the phase shift system takesplace according to the following table:

V₁₁ V₁₂ V₂₁ V₂₂ shift −5 V −5 V −5 V −5 V 0°   5 V   5 V −5 V −5 V +120°−5 V −5 V   5 V   5 V +120°   5 V   5 V   5 V   5 V −120°

The other combinations are not commonly used.

7.2.4. Beam deflector

The beam deflector is represented in FIGS. 37 and 38. Its basic unit isan elementary variation block made up of the elements (2801) to (2804).

An elementary variation block is made up of a first rotator (2801)denoted “R1” followed by a birefringent prism (2802) denoted “PD”(deviation prism) then a second rotator (2803) denoted “R2” and aGlan-Thomson polarizer (2804) denoted “POL”. The rotator (2801) has itsneutral axis in the vertical direction for an applied voltage of −5 V.For the same applied voltage, the rotator (2803) has its neutral axis inthe horizontal direction. The prism (2802) is made up of a birefringentmaterial, calcite or quartz. The polarization direction of theextraordinary ray (first neutral axis) is vertical for example, and thepolarization direction of the ordinary ray (second neutral axis) ishorizontal. An incident ray on this prism is thus divided into anordinary ray polarized in the vertical direction and an extraordinaryray polarized in the horizontal direction. The ordinary ray and theextraordinary ray have a different inclination at the exit of the prism(angle of their propagation direction with that of the incoming beam).

FIGS. 39 and 40 illustrate the functioning of this elementary variationblock. FIG. 39 corresponds to a deflection in one direction and FIG. 40to a deflection in the other direction. The arrows in bold represent theelectric field vectors of the considered beams.

In the case of FIG. 39, the voltages applied to the two rotators arerespectively −5 V for (2801) and +5 V for (2803). The electric fieldvector of the incoming beam is vertical (2901). After passing throughthe first rotator whose neutral axis (2902) is vertical, it remainsvertical (2903). After passing through the deflection prism, it is madeup of a single extraordinary ray (2904). After passing through thesecond rotator, it is symmetrized in relation to the axis (2906) of thisrotator, which itself forms an angle of 40 degrees with the horizontal(it is assumed for the drawing that α=40° but the result does not dependon the exactitude of this value). It is thus transformed into a vector(2905) forming an angle of 10 degrees with the horizontal. The polarizerprojects this vector on to the horizontal to obtain the vector (2907)whose deflection corresponds to only the extraordinary ray.

In the case of FIG. 40, the voltages applied to the two rotators arerespectively +5 V for (2801) and −5 V for (2803). The field vector ofthe incoming beam is vertical (2911). After passing through the firstrotator, it is symmetrized in relation to the axis (2912) of thisrotator, which itself forms an angle of 40° with the vertical. It isthus transformed into a vector (2913) forming an angle of 10 degreeswith the horizontal. After passing through the deflection prism, thebeam is split into an extraordinary beam having a field vector (2914)and an ordinary beam having a field vector (2915). After passing throughthe second rotator, with horizontal axis, the field vector of theextraordinary beam is symmetrized in relation to the horizontal andbecomes (2916). The polarizer then selects only the horizontal componentand the outgoing vector (2917) thus corresponds to the ordinary rayalone.

A complete elementary block is represented by the rectangle (2805), thedirection of the field of the incoming beam being represented by thearrow (2806). The block (2807) is identical but turned 90 degrees inrelation to the horizontal axis so that the direction of the field ofthe incoming beam is horizontal (2808). Both blocks give an elementarydoublet (2809) allowing an elementary deflection of the beam in thehorizontal and vertical directions. As indicated in FIG. 38, the entiredeflector is made up of eight successive elementary doublets. However,in order to have an efficient switching system, the last doublet(numbered 0) is placed on the part of the beam in which the left-handand right-hand illuminating beams have already been separated. Twoidentical doublets (D0) and (D0 b) are thus used, one on each branch ofthe beam. When a voltage of −5 V is applied to its two rotators, anelementary block acts as a closed switch. The last doublet can thusswitch the beam effectively, a voltage of −5 V having to be applied toall its rotators to have a closed switch.

The block (2209) in FIG. 27 thus represents the doublet D0 b. The block(2240) represents the doublet D0. The block (2206) represents thedoublets D1 to D7.

The type of crystal in which the prism is manufactured and the anglebetween its two faces determine the variation angle of the beam'sinclination between the two positions of an elementary doublet.

The following notations are adopted:

n_(l): index of the immersion liquid used for the beam deflectiondevice.

o: aperture of the objective.

A: aperture of the beam at the entrance of the deflection device,roughly equal to its value at the level of the diaphragm where the imageis formed. A is defined as the sine of a half-angle at the vertex of thecone formed by the rays coming from the objective.

g: magnification of the objective.

d: distance between planes P2 and P4 of FIG. 29.

f₁: focal length of the lens L1 in FIG. 29.

Abbe's formula and the solution of the optical equations yields:$A = {\frac{o}{n_{1}\quad g}\quad \left( {1 - \frac{d}{f_{1}}} \right)}$

FIG. 41 shows the calculation principle for the deflection of ordinaryrays by the prism. The ray (2922) enters the prism at (2921) and comesout at (2923).

θ_(d) is the angle at the vertex of the deflection prism.

θ_(e) is the angle of the exiting extraordinary beam with the externalface of the prism.

θ_(o) is the angle of the exiting ordinary beam with the external faceof the prism.

n_(e) is the extraordinary index of the deflection prism.

n_(o) is the ordinary index of the deflection prism.

We have:${\sin \quad \theta_{o}} = {\frac{n_{o}}{n_{l}}\quad \sin \quad \theta_{d}}$

and likewise for the extraordinary ray:${\sin \quad \theta_{e}} = {\frac{n_{e}}{n_{l}}\quad \sin \quad \theta_{d}}$

whence:${{\sin \quad \theta_{e}} - {\sin \quad \theta_{o}}} = {\frac{n_{e} - n_{o}}{n_{l}}\quad \sin \quad \theta_{d}}$

or, in the first order in (θ_(e)−θ_(o)):${\left( {\theta_{e} - \theta_{o}} \right)\quad \cos \quad \theta_{d}} = {\frac{n_{e} - n_{o}}{n_{l}}\quad \sin \quad \theta_{d}}$

The i-th doublet must create a variation in the inclination ofamplitude: ${\theta_{e} - \theta_{o}} = \frac{A}{2^{i}}$

The half-angle at the vertex of the i-th prism is thus equal to:$\theta_{d} = {{Arctan}\quad \left( {\frac{A}{2^{i}}\quad \frac{n_{l}}{n_{e} - n_{o}}} \right)}$

or with the value of A previously obtained:$\theta_{d} = {{Arctan}\quad \left( {\frac{1}{2^{i}}\quad \frac{1}{g}\quad \frac{o}{n_{e} - n_{o}}\quad \left( {1 - \frac{d}{f_{1}}} \right)} \right)}$

In this equation, the following values must be taken into account:

quartz: n_(e)−n_(o)=0.009

calcite: n_(e)−n_(o)=−0.172

For each prism:

Inverting the ordinary and extraordinary axes makes it possible toinvert the direction in which the rays are deflected when changing froma pair of voltages (−5 V, 5 V) to a pair of voltages (5 V, −5 V) appliedto the rotators of the elementary block concerned. As this direction isinverted between the quartz and the calcite for a given choice ofordinary and extraordinary axes, these axes must be inverted in acalcite prism in relation to their position in a quartz prism.

Inverting the orientation of the prism (vertex oriented downward insteadof upward) also makes it possible to invert the direction in which therays are deflected when changing from a pair of voltages (−5 V,5 V) to apair of voltages (5 V, −5 V) applied to the rotators of the elementaryblock concerned. However, at the same time, this operation inverts thedirection of deflection of the rays when a fixed pair of voltages isapplied. In order to have a “fixed” minimum deflection of the rays, theprism with the greatest deflection of each series, calcite or quartz,must be inverted in relation to the others. In order to maintain thedeflection in the desired direction, its ordinary and extraordinary axesmust be inverted.

For each prism is chosen the material, quartz or calcite, which makes itpossible to obtain most easily this angle at the vertex. The orientationof the prism is then chosen so that, at a fixed voltage applied to therotators, the direction variations induced are best compensated betweenthe prisms. The position of the ordinary and extraordinary axes arechosen so that the rays are always deflected in the same direction whenchanging from a pair of voltages (−5 V, 5 V) to a pair of voltages (5 V,−5 V) applied to the rotators of the elementary block concerned. Foreach doublet, it is necessary to specify for the two prisms included inthe doublet, which have the same characteristics: the angle at thevertex, the orientation of the vertex (normal or inverted in relation toFIG. 37 in which it is oriented upward), and the position of theordinary or extraordinary axes (normal or inverted in relation to FIG.37 in which the extraordinary axis is vertical). For example, for anobjective o=1.25 g=100 and with f₁=200 mm, d=20 mm, the following tableis obtained in which the angles are in degrees:

Position of ordinary and Index of Orientation of extraordinary prismθ_(d) (calcite) θ_(d) (quartz) Choice vertex axes 0 3.742 51.340 calciteinverted normal 1 1.873 32.005 calcite normal inverted 2 0.937 17.354calcite normal inverted 3 0.468 8.881 quartz inverted inverted 4 0.2344.467 quartz normal normal 5 0.117 2.237 quartz normal normal 6 0.0581.119 quartz normal normal 7 0.029 0.559 quartz normal normal

A deflection of the beam is a variation in its direction. However, at agreat distance from the doublet creating the deflection, it also resultsin a spatial shift of the illuminated area. In order for this phenomenonnot to be penalizing, the distances between the elements of thedeflector must be reduced to a minimum and these elements must have asufficient section so that, whatever the chosen orientation, the beam“completely fills” the area delimited by the diaphragm. For example,this section may be 12 mm, the Glan-Thomson polarizers then having thedimension 12×30 mm. All the parts of the deflector which do not transmitthe beam directly must be as absorbing as possible in order to limitnoise.

To suppress the constant deflections by the deflection prisms, the indexof the optical liquid in which a prism is immersed must be equal to theaverage value of the ordinary and extraordinary indices of the prism,i.e.:${{For}\quad {{calcite}:n_{l}}} = {\frac{1.658 + 1.486}{2} = 1.572}$${{For}\quad {{quartz}:n_{l}}} = {\frac{1.544 + 1.553}{2} = 1.5485}$

The part of the beam deflector whose prisms are in calcite must hence beimmersed in a liquid of 1.572 index, and the part whose prisms are inquartz must be immersed in a liquid of 1.5485 index. The recipientcontaining the deflector and the optical liquid must consequently beseparated into two parts, with a glass window allowing the passage ofthe beam between these two parts which contain optical liquids ofdifferent index.

The device is controlled by controlling the 36 rotators. In eachdoublet, the phase rotators are numbered from 0 to 3, number 0 being the“leftmost” rotator in FIG. 28. If i is the index of the doublet, varyingfrom 0 to 7, and j is the index of the rotator in a doublet, varyingfrom 0 to 3, the overall index k=i+1+j*9 is then assigned to therotator, except for the doublet numbered 0b for which we have k=j*9. A36-bit control word is used in which the bit number k corresponds to therotator of overall index k. For each bit, a value of 0 corresponds to anapplied voltage of −5 V, and a value of 1 corresponds to a voltage of +5V.

An illumination is characterized by the sensor on which the directilluminating beam arrives and by symbolic coordinates on this sensor.The sensor will be indexed by the integer p and the symbolic coordinateswill be i,j where i and j vary between 0 and 255. The symboliccoordinates do not correspond necessarily to coordinates in pixels onthe sensor. When one wishes to obtain an illumination characterized bythe indices p,i,j, the control word is given in the following table:

Coor- Sensor dinates Control word COM[p,i,j] (2239): (i,j)(0₁,i₈,0₁,{overscore (i)}₈0₁,j₈,0₁,{overscore (j)}₈) p = 0 (2229): (i,j)(i%2)₁,0₁,(i/2)₇,{overscore ((i%2))}₁,0₁,{overscore ((i/2))}₇,(j%2)₁,0₁,(j/2)₇, p = 1 {overscore ((j%2))}₁,0₁,{overscore ((j/2))}₇)

In this table:

a%b means a modulo b

a/2 represents the integer part of the division of a by 2, i.e. ashifted to the right

(a,b,c . . . ) represents a concatenated with b then with c etc. a_(i)represents a expressed over i bits.

If a is an integer number, its expression in binary is a series of 0and 1. By transforming the 0's to 1's and vice versa, we obtain itscomplement which is denoted {overscore (a)}. This notation will bemaintained hereinafter.

When one wishes to suppress the two illuminating beams, the control wordto be used is 0.

7.3. Overall adjustment

7.3.1. First setup of the system

The system, with the exception of the elements (2204) (2205) (2206)(2209) (2240) (2210) (2241) (2238) (2226), is put in place geometricallywith maximum precision. The path of the beam is checked by using adiffusing piece of paper inserted along its path. The position of themirrors, semi-transparent mirrors, as well as of (2200) and (2201), isadjusted by thus checking the path of the beam.

7.3.2. Setup of beam control systems

For the setup of the beam control systems, it is necessary to have aphotometer of sufficient accuracy, which will be used to measure beamattenuations or detect extinctions. These systems are composed ofoptical elements (prisms, rotators, polarizers, birefringent plates)which must be positioned accurately in relation to the optical axis andbe mounted on positioners allowing fine rotational adjustment aroundthis axis.

7.3.2.1. Marking of rotators

The entire illuminating beam modification system is based on the use ofphase rotators. It is essential for the axis of each rotator to turn inthe direction planned upon the application of a voltage opposite to thatused during its setup. The position of the rotator must be defined whenit is put in place and the adjustment takes place only on a few degrees.To specify the position of the axes of the rotator before setup, a testbetween crossed polarizers is carried out, in two stages.

Stage 1, described in FIG. 42, the rotator (3001) is put in placebetween the entrance polarizer having a polarization direction (3002),and the exit polarizer having a polarization direction (3003). A voltageof +5 V is applied. The rotator is adjusted in rotation so as to cancelthe outgoing ray. The corresponding position, corresponding to theentrance polarizer, is marked with a red point (3004).

Stage 2: A voltage −5 V is applied. The exit polarizer is adjusted tocancel the ray. A green point is marked corresponding to the middle ofthe two polarizer positions, on the side where the angle is smallest.FIGS. 43 and 44 describe this stage in the two possible cases defined bythe new position of the exit polarizer, respectively (3005) and (3006).In the case of FIG. 43, the green point is marked at (3007) and in thecase of FIG. 44 it is marked at (3008).

The red point then marks the position of the axis for a +5 V voltage andthe green spot marks its position for a −5 V voltage. These points thenallow the correct prepositioning of the elements during the adjustmentprocedure.

7.3.2.2. Setup of beam attenuator

The polarizer (2502) is first put in place and a fine adjustment inrotation is carried out in order to reach the extinction of the outgoingbeam. The rotator (2501) is then put in place and a fine adjustment ofits position in rotation is carried out in order to have the attenuationa₁ ² sought when changing from the open position (+5 V voltage appliedto (2501)) to the closed position (−5 V voltage applied).

We then switch (2501) to the open position (+5 V voltage) and thepolarizer (2504) is put in place and adjusted in rotation in order toreach the extinction of the outgoing beam. The rotator (2503) is thenput in place and its position in rotation is fine-adjusted in order tohave the attenuation a₂ ² sought when changing from the open position(+5 V voltage applied to (2503)) to the closed position (−5 V voltageapplied). For example, it is possible to use${a_{1} = {a_{2} = \frac{1}{16}}},$

which gives attenuations measurable by the photometer:$a_{1}^{2} = {a_{2}^{2} = \frac{1}{256}}$

The exact values obtained for the coefficients a₁ ² and a₂ ² are thenmeasured. The coefficients a₁ and a₂ thus obtained will be used later.

7.3.2.3. Setup of phase shift devices

These devices (2205) and (2251) are put in place with the best possibleprecision considering the marks made previously.

7.3.2.4. Setup of the beam deflection and switching device

Each elementary block is put in place successively beginning with theblock closest to the phase shift device. The beam attenuator must be inthe open position. An elementary block is put in place in the followingorder:

Placement of polarizer POL (2804). Fine adjustment in rotation to cancelthe exiting ray.

Placement of rotator R2 (2803). Fine adjustment in rotation to keep theexiting ray at a zero value, a −5 V voltage being applied to R2.

Placement of rotator R1(2801). Fine adjustment in rotation to keep theexiting ray at a zero value, a −5 V voltage being applied to R2 and R1.

Placement of deflection prism PD (2802). Fine adjustment in rotation tokeep the exiting ray at a zero value, a −5 V voltage being applied to R1and R2.

The blocks D0 and D0 bare put in place in the same manner as the others.

7.3.2.5. Setup of phase rotators (2210) (2241) (2238) (2226)

These rotators must have their neutral axis vertical for an appliedvoltage of −5 V (green point up). The axis of (2210) must turn to theright in FIG. 27 when a +5 V voltage is applied (red point to theright). The axis of (2241) must turn to the left in FIG. 27 when a +5 Vvoltage is applied (red point to the left). The axis of (2238) and(2226) must turn up in FIG. 27 when a +5 V voltage is applied (red pointup).

7.3.2.6. Setup of polarizers

The polarizers (2252) and (2253) are put in place with their passingaxis vertical.

7.3.3. Adjustment of geometry

In a second phase, a geometry adjustment is carried out to correctlyposition the cameras, the achromatic lenses and certain mirrors. Some ofthese adjustments use an auxiliary CCD sensor whose pitch (distancebetween the centers of two adjacent pixels) must be as small aspossible. From the image received either on one of the system sensors,or on the auxiliary sensor, an algorithm is used that allows theevaluation of the punctuality of the image and the location of themaximum. The image received on a sensor is obtained by causing theinterference on this sensor of a reference wave and the wave whosepunctuality is to be evaluated.

7.3.3.1. Obtaining a two-dimensional image and evaluation of punctuality

It was seen in the first embodiment how a two-dimensional image incomplex numbers can be generated from three images differing from eachother in the phase difference between the illuminating wave and thereference wave. This image in the frequency domain can be transposed tothe space domain by Fourier transform. In the adjustment phases thatfollow, the evaluation of the point nature (punctuality) and thecentering of such images in the spatial or frequency domain will becalled for. These images will either be received on one of the CCDsensors of the device or on an auxiliary sensor. The phase shifts willbe obtained either by means of (2205) or by means of (2251). In certaincases, the illuminating wave will play the role of the reference waveand vice versa. In every case, the system consequently produces threesuccessive images with successive shifts indexed by the integer d, of+120 (d=0), 0(d=1), −120(d=2) degrees. This gives an array of pixelsI[d,i,j] in which the index d varying from 0 to 2 indexes the phasedifference. The image in complex numbers is deduced therefrom by:${S\left\lbrack {i,j} \right\rbrack} = \left\lbrack \left. {{\frac{1}{6}\left( {{2{I\left\lbrack {0,i,j} \right\rbrack}} - {I\left\lbrack {1,i,j} \right\rbrack} - {I\left\lbrack {2,i,j} \right\rbrack}} \right)} + {\overset{\sim}{j}\frac{1}{2\sqrt{3}}\left( {{I\left\lbrack {1,i,j} \right\rbrack} - {I\left\lbrack {2,i,j} \right\rbrack}} \right)}} \right\rbrack \right.$

if one seeks to evaluate punctuality in the space domain, inverseFourier transformation is applied to this image.

In both cases (whether Fourier transformation has taken place or not),from this image having the dimensions of N_(pix)×N_(pix) correspondingto the number of useful pixels of the sensor concerned (for example256), punctuality is evaluated by the program, comprising the followingstages:

Stage 1: The program calculates the maximum max of the modulus of S[i,j]and determines its coordinates (imax1,jmax1).

Stage 2: The part of the array S[i,j] located around (imax1,jmax1) isextracted. One thus creates an array Sa[i,j] having the dimensionsN_(a)×N_(a) where, for example N_(a)=16:${{Sa}\left\lbrack {i,j} \right\rbrack} = {S\left\lbrack {{i - \frac{N_{a}}{2} + {imax1}},{j - \frac{N_{a}}{2} + {jmax1}}} \right\rbrack}$

Stage 3: A direct Fourier transform is taken on array Sa

Stage 4: Array Sa is completed by zeros and an array Sb with dimensionsof N_(b)×N_(b) is obtained${{Sb}\left\lbrack {i,j} \right\rbrack} = {{Sa}\left\lbrack {{i - \frac{N_{b}}{2} + \frac{N_{a}}{2}},{j - \frac{N_{b}}{2} + \frac{N_{a}}{2}}} \right\rbrack}$

 when${N_{a} - 1} \geq {i - \frac{N_{b}}{2} + \frac{N_{a}}{2}} \geq {{0\quad {and}\quad N_{a}} - 1} \geq {j - \frac{N_{b}}{2} + \frac{N_{a}}{2}} \geq 0$

 and Sb[i,j]=0 otherwise

Stage 5: The inverse Fourier transform of array Sb is taken. One thusobtains an oversampled version of the part of the initial array Slocated around the point corresponding to the modulus maximum value.

Stage 6: imax,jmax,max are calculated by the formulas:${imax2} = \frac{\sum\limits_{i,j}\quad {{{{Sb}\left\lbrack {i,j} \right\rbrack}}^{2}i}}{\sum\limits_{i,j}\quad {{{Sb}\left\lbrack {i,j} \right\rbrack}}^{2}}$${jmax2} = \frac{\sum\limits_{i,j}\quad {{{{Sb}\left\lbrack {i,j} \right\rbrack}}^{2}j}}{\sum\limits_{i,j}\quad {{{Sb}\left\lbrack {i,j} \right\rbrack}}^{2}}$max  = Sb[imax2, jmax2]${imax} = {{imax1} + {\frac{N_{a}}{N_{b}}\left( {{imax2} - \frac{N_{b}}{2}} \right)}}$${jmax} = {{jmax1} + {\frac{N_{a}}{N_{b}}\left( {{jmax2} - \frac{N_{b}}{2}} \right)}}$

The real values max,imax,jmax thus obtained characterize respectivelythe value and the position of the maximum. The higher the value max thebetter the punctuality. The program also displays the modulus of array Sand the modulus of array Sb to have a visual evaluation of punctuality.

7.3.3.2. Apparatus used

A diffuser is used, for example a piece of paper allowing the path ofthe beam to be followed visually.

An auxiliary CCD is used to follow the beam more precisely than with thediffuser. Its pitch must be as small as possible.

A frequency-meter is also used. This term will designate the instrumentdescribed in FIG. 71, used to measure the spatial frequencies of aparaxial beam. It is made up of a mirror (5000) which reflects anincoming parallel beam towards a lens (5001) which focuses this beamtowards a CCD (5002) mounted on a camera (5003). An optional polarizer(5004) can be inserted between the mirror and the lens. The use of themirror (5000) allows the size of the frequency-meter to be kept to aminimum in the horizontal plane which is that of FIGS. 61 and 62. Theoptical axis of the lens and of the CCD is always vertical duringmeasurement operations.

Given the dimensioning choices made, the maximum angle under which thebeams enter the frequency-meter is o/g. If the total width of the CCD(5002) is 1, then the focal length of (5001) is calculated for the raysarriving under angles between −o/g and o/g to be taken into account. Itis thus equal to: $f = {\frac{g}{o}{\frac{1}{2}.}}$

A slightly lower value may be adopted to maintain a safety margin.

Before using the frequency-meter, the distance between the lens (5001)and the CCD (5002) must be adjusted so that the image of a parallel beamis as near a point image as possible, which is accomplished simply bysending to the frequency-meter a beam whose parallelism has beenpreviously checked by the interferometry method and by adjustingaccordingly the distance between the lens and the CCD.

The punctuality of the image obtained on the CCD (5002) allows thechecking of the parallelism of an incoming beam. The relative positionof several points on this CCD characterizes the angle between thecorresponding beams.

Unless otherwise indicated in the description of an adjustment, thepolarizer (5004) is not used.

7.3.3.3. Adjustment cycle

Adjustments are designed to ensure that:

(1) The beams follow the planned path. This can generally be checked bymeans of a simple diffuser. These beam path checks are not described butmust be carried out prior to the other adjustments. For example, theorientation of the mirror (2247) must be adjusted so that the reflectedbeam is in fact superimposed on the incident beam, occupying the samespace zone as the latter at the level of the semi-transparent mirror(2245).

(2) The illuminating beams and their opposite indicators have a pointimage on the CCD sensors.

(3) The reference beams have a point image in the plane of thediaphragms (2213) and (2223).

(4) A parallel beam entering the objective (2217) and directed along theoptical axis has a point image centered on the sensor (2239).

(5) When a control word COM[1,i,j] is used, the coordinates of the pointilluminated by the beam FED are deduced from those of the pointilluminated by FEDI by homothetic similitude with a ratio close to 1.

The adjustments to be carried out result from these conditions. Thedescription of the adjustment stages is given by way of information andconstitutes an example of the ordering of the adjustment stages.

Throughout the adjustment, with the exception of Stage 14, the controlword used for the beam deflector is COM[1,128,128], COM[0,128,128] or 0depending on whether the right-hand illuminating beam FED or theleft-hand illuminating beam FEG, or neither, is generated. The shutters(2257) (2258) (2259) (2260), and shutters not represented on the path ofbeams FRG and FRD, allow the selection of the beams used.

During certain adjustment phases, a given beam is measured on a sensorby means of a second beam serving as a reference. The program describedin 7.3.3.1. is then used to evaluate the punctuality of the measuredbeam. Phase variations between the beam serving as a reference and thebeam to be measured are obtained by means of (2205) or (2251). When nobeam has been used as a reference, for example if the CCD is that of thefrequency-meter, the image used is that received directly on the CCD. Apoint image is considered to be centered on a sensor of sizeN_(pix)×N_(pix) if its coordinates are$\left( {\frac{N_{pix}}{2},\frac{N_{pix}}{2}} \right).$

Stage 1: Lens (2231) Translation Adjustment

The frequency-meter is positioned between the semi-transparent mirror(2211) and the polarization rotator (2238). The lens (2231) is adjustedso that the image of the beam FEDI on the CCD of the frequency-meter isa point.

Stage 2: Lens (2246) Translation Adjustment

The frequency-meter is positioned between the semi-transparent mirror(2245) and the lens (2237). The lens (2246) is adjusted so that theimage of the beam FRDI on the CCD of the frequency-meter is a point.

Stage 3: Lens (2242) Translation Adjustment

The frequency-meter is positioned between the semi-transparent mirror(2225) and polarization rotator (2226). The lens (2242) is adjusted sothat the image of the beam FEGI on the CCD of the frequency-meter is apoint.

Stage 4: Lens (2281) Translation Adjustment

The frequency-meter is positioned between the semi-transparent mirror(2228) and the lens (2227). The lens (2281) is adjusted so that theimage of the beam FRGI on the CCD of the frequency-meter is a point.

Stage 5: Lens (2212) Translation Adjustment

A provisional illuminating beam FEP is introduced. It is deriveddirectly from the output of the beam expander (2201) by means of asemi-transparent mirror and is redirected by a set of mirrors to theobjective (2217) in which it penetrates through the side where thesample is normally located, and directed along the optical axis of theobjective. The objective (2217) must be roughly in a focussed position,i.e. in the position in which it will be during the normal use of themicroscope. The objective (2219) must be temporarily eliminated in orderto be able to introduce FEP.

The frequency-meter is positioned between the lens (2212) and thesemi-transparent mirror (2211). The lens (2212) is adjusted so that theimage of the beam FEP on the CCD of the frequency-meter is a point.

Stage 6: Lens (2237) Translation Adjustment and Semi-transparent Mirror(2245) Orientation Adjustment

An auxiliary CCD is placed in the location of the diaphragm (2213). Thelens (2237) is adjusted so that the image of the beam FRDI on thisauxiliary CCD is a point. The semi-transparent mirror (2245) is adjustedso that the image of the beam FRDI on this auxiliary CCD is centered.

Stage 7: CCD (2239) Translation Adjustment

The position of the CCD (2239) is adjusted so that the image of the beamFEP obtained using the procedure described in 7.3.3.1. from the CCD(2239) is a centered point image.

Stage 8: Semi-transparent Mirror (2211) Adjustment

The position of this mirror is adjusted so that the image of the beamFEDI obtained using the procedure described in 7.3.3.1. from the CCD(2239) is a centered point image.

Stage 9: Objective Position Adjustment

The beam FEP is eliminated and the objective (2219) is put in placeagain. An auxiliary CCD is placed at the location of the diaphragm(2223). The sample is a transparent plate, for example, and optical oilis used on each side of the plate. The position of the objectives isadjusted so that the image of the beam FRDI on this auxiliary CCD is acentered point image.

Stage 10: Lens (2224) Translation Adjustment

The frequency-meter is positioned between the lens (2224) and thesemi-transparent mirror (2225). The lens (2224) is adjusted so that theimage of the beam FED on the CCD of the frequency-meter is a point.

Stage 11: Lens (2227) Translation Adjustment and Semi-transparent Mirror(2228) Orientation Adjustment

An auxiliary CCD is placed at the location of (2223). The lens (2227) isadjusted so that the image of the beam FRGI on this auxiliary CCD is apoint. The semi-transparent mirror (2228) is adjusted so that the imageof the beam FRGI on this auxiliary CCD is centered.

Stage 12: CCD (2229) Translation Adjustment

The position of the CCD (2229) is adjusted so that the image of the beamFED obtained using the procedure described in 7.3.3.1. from the CCD(2229) is a centered point image.

Stage 13: Semi-transparent Mirror (2225) Adjustment

The position of this mirror is adjusted so that the image of the beamFEGI obtained using the procedure described in 7.3.3.1. from the CCD(2229) is a centered point image.

Stage 14: CCD (2229) and (2239) Rotation and Translation Adjustment

This stage consists in adjusting the position of (2229) and (2239) inrotation so that their systems of axes coincide. For this purpose, thecontrol words COM[1,128,128], COM[1,250,128], COM[1,128,250] are usedalternately. The two sensors are adjusted in translation in a planeorthogonal to the axis (2264) and in rotation around this same axis. Oneach sensor is defined a system of coordinates (i,j) in which the pixelindices i and j range from 0 to N_(pix)−1 where N_(pix)=256, and whichdetermines coordinates in the useful zone of the sensor, which will beused hereinbelow. The origin of the system of coordinates, namely thecoordinates (0,0), may be any of the four corners of the useful zone ofthe sensor. At the same time as this adjustment is carried out, theorigin of the system of coordinates is chosen. The adjustment and originselection criteria are the same for the two sensors and are as follows:

When COM[1,128,128] is used, the point of impact of the illuminatingbeam, i.e. the direct illuminating beam on (2229) or the oppositeindicator on (2239), must be at the coordinates (128,128).

When COM[1,250,128] is used, the point of impact of the illuminatingbeam must be at a point of coordinates (x,128) where x is positive.

When COM[1,128,6] is used, the point of impact of the illuminating beammust be a point of coordinates (128,x) where x is positive.

The coordinate systems thus determined are defined by their origin O_(p)and their unit base vectors {right arrow over (i)}_(p),{right arrow over(j)}_(p), i.e. by the references (O_(p),{right arrow over(i)}_(p),{right arrow over (j)}_(p)) where p is the index of the sensor(p=0 for (2239) and p=1 for (2229)).

This system of coordinates will be systematically used hereinbelow.

7.4. Determining the reference wave modulus

As in the first embodiment, the filters (2255) and (2256) are adjustedso that the level of the reference wave is about one-fourth the maximumlevel authorized by the digitizer, i.e. a level of 256/4=64 in the caseof 8-bit sampling of the video signal, and this on both sensors.

This reference wave is then determined as in the first embodiment, butsince both sensors are present, an array Iref[p,i,j] is obtained inwhich i,j are the pixel indices as in the first embodiment, and in whichp is the sensor index, i.e. p=0 for (2239) and p=1 for (2229). Iref[0,i,j] is the intensity received on the sensor (2239) when only the beam FRDis present, and Iref[1,i,j] is the intensity received on the sensor(2229) when only the beam FRG is present.

7.5. Simple two-dimensional imaging

It was seen in the first embodiment how a two-dimensional frequencyrepresentation in complex numbers can be generated from three imagesdiffering from each other in the phase difference between theilluminating wave and reference wave. In the adjustment phases tofollow, this type of frequency representation will be generated. Togenerate such a representation, the system produces three successiveimages with successive shifts of +120, 0,−120 degrees applied to theilluminating wave. The order of the phase differences is inverted inrelation to that used in the first embodiment because they are appliedto the illuminating wave and not to the reference wave. This yields anarray of pixels I[d,p,i,j] in which the index d, varying from 0 to 2,designates the phase difference and in which the index p designates thesensor. The frequency representation is deduced therefrom by:${S\left\lbrack {p,i,j} \right\rbrack} = \left\lbrack {{\frac{1}{6\sqrt{{Iref}\left\lbrack {p,i,j} \right\rbrack}}\left( {{2{I\left\lbrack {0,p,i,j} \right\rbrack}} - {I\left\lbrack {1,p,i,j} \right\rbrack} - {I\left\lbrack {2,p,i,j} \right\rbrack}} \right)} + {\overset{\sim}{j}\frac{1}{2\sqrt{3{{Iref}\left\lbrack {p,i,j} \right\rbrack}}}\left( {{I\left\lbrack {1,p,i,j} \right\rbrack} - {I\left\lbrack {2,p,i,j} \right\rbrack}} \right)}} \right\rbrack$

With each acquisition is obtained an image for each sensor, and thearray S[p,i,j] thus includes a sub-array for each sensor, the index pdesignating the sensor. However, in the adjustment phase, only one ofthese two images will generally be used.

7.6. Obtaining Parameters K_(p) and adjustment of diaphragms

These parameters correspond to the parameter K of the first embodiment,but because of the dissymmetry of realization, the parameter is notnecessarily the same for each sensor. Two parameters K₀ and K₁ willconsequently be defined, corresponding to the two sensors. Theseparameters are obtained as in the first embodiment by simple imagingusing an objective micrometer. The objective micrometer used musthowever be designed for this purpose: the marks must be made on a finecover glass, as a thick plate cannot be used with this microscope.

In a first step, the objectives must be correctly focussed. For thispurpose, the parallel beams FED and FEG are eliminated. Only thereference waves FRD and FRG are present. The mirrors (2282) and (2247)are used to obtain centered waves with two propagation directions. Themirrors (2243) and (2232) are blocked. The phase shifter (2251) is usedto modify the phase of the waves. The micrometer is moved so that themarks are outside the field of the objectives. The wave measured on oneside of the objectives is the equivalent of the reference wave used onthe other side. The program described in 7.3.3.1. with Fouriertransformation allowing passage from the frequency domain to the spatialdomain, is used to evaluate the point nature of the image. The positionof the objectives is adjusted to obtain a point centered in the middleof the image.

In a second step, the sample must be correctly positioned. The mirrors(2282) (2247) (2243) (2232) are blocked. Only the beam FEG is used, witha beam deflector control word COM[0,128,128]. The position of the samplein the horizontal plane is modified until a modification is obtainedwhich is characteristic of the wave received on the sensors in theabsence of a reference wave, as in the first embodiment.

The reference wave FRD is then reintroduced and a focussing programsimilar to that used in the first embodiment, and using the imageobtained on the sensor (2239), is started. This program differs howeverfrom the one used in the first embodiment in that the illuminating wavephase and amplitude modifications are now controlled by the devices(2204) and (2205) and in that the value rapport_moy characteristic ofthe attenuation is real and is that which was measured in 7.3.2.2.between the only two positions of the beam attenuator which are usedhere.

The position of the diaphragm (2213) is then adjusted in a planeperpendicular to (2264) so that its image is correctly centered. Itsaperture is adjusted in order to be as high as possible; the diaphragmmust however remain entirely visible on the obtained image. The positionof the micrometer is then adjusted in the vertical direction in order toobtain a clear image. On this image, the distance (D_(pix))₀ between twograduations separated by a real distance D_(real) is measured.

The beams FEG and FRRD are then eliminated. The beam FED is introducedwith the beam deflector control word COM[1,128,128]. The reference beamFRG is also introduced. The focussing program is restarted, this timeusing the image received on the sensor (2229). The position of thediaphragm (2223) is then adjusted in a plane perpendicular to (2264) sothat its image is correctly centered. Its aperture is adjusted in orderto be as high as possible; the diaphragm must however remain entirelyvisible on the obtained image. On the obtained image is measured thedistance (D_(pix))₁ between the same graduations as previously.

We then have$K_{p} = {\frac{n_{v}}{\lambda}\frac{N_{pix}}{\left( D_{pix} \right)_{p}}D_{real}}$

where:

(D_(pix))_(p) is the distance in pixels measured on the image comingfrom the sensor index p, where p=0 for (2239) and p=1 for (2229)

D_(real) is the same for each measurement.

N_(pix) is the number of pixels of the useful zone of the sensor and ofthe Fourier transform used in the focussing program, for exampleN_(pix)=256.

The diaphragm apertures obtained after this procedure will be maintainedin the rest of the description.

7.7. Path difference induced on a parallel beam

The average index of the observed sample is not necessarily the nominalindex of the objectives. This index difference can lead to significantspherical aberration. It is necessary, to correct this aberration, totake into account the average index n_(o) in the sample and thethickness L of the sample between two cover glasses with the nominalindex.

The subsequent reconstitution of a three-dimensional image also requiresknowledge of the relative position of the virtual origins of thereference beams used on each side of the system. This relative positionis defined by x,y,z which are the coordinates of the origin of thereference beam used in the left-hand part in relation to the origin ofthe beam used in the right-hand part.

The parameters x,y,z,L, n_(o) lead to phase differences in a parallelwave passing through the sample. The calculation of these phasedifferences is set forth below:

Correspondence between the direction of a parallel beam in the objectand the coordinates of the corresponding point on the sensor:

i and j are the coordinates of a pixel in relation to the opticalcenter, expressed in pixels, on the sensor p, in the system ofcoordinates defined in Stage 14 of the adjustment cycle described in7.3.3.2. A unit vector parallel to the frequency vector in the objecthas the coordinates, in an orthonormed coordinate system {right arrowover (a)}_(p),{right arrow over (b)}_(p), {right arrow over (c)}_(p) inwhich {right arrow over (c)}_(p) is parallel to (2263) and in which{right arrow over (a)}_(p),{right arrow over (b)}_(p) is deduced simplyfrom the vectors {right arrow over (i)}_(p),{right arrow over (j)}_(p)defined in 7.3.3.3.:$u = \left( {\frac{i}{K_{p}},\frac{j}{K_{p}},{\frac{1}{K_{p}}\sqrt{K_{p}^{2} - i^{2} - j^{2}}}} \right)$

 and it forms with the axis (2263) and angle of:${\sin \quad \alpha} = {\frac{1}{K_{p}}\sqrt{i^{2} + j^{2}}}$

If the beam coming from the objective were not deflected by the mirrors(2214) (2215) (2216) or (2222) (2221) (2220), the vectors {right arrowover (a)}_(p),{right arrow over (b)}_(p) would be equal to the vectors{right arrow over (i)}_(p),{right arrow over (j)}_(p). Although this isnot really the case, the basic vectors {right arrow over (a)}_(p),{rightarrow over (b)}_(p),{right arrow over (c)}_(p) will be denotedhereinafter by {right arrow over (i)}_(p),{right arrow over(j)}_(p),{right arrow over (k)}_(p), without differentiating the vectors{right arrow over (a)}_(p),{right arrow over (b)}_(p) from thecorresponding vectors {right arrow over (i)}_(p),{right arrow over(j)}_(p) defined in 7.3.3.3. on the sensors.

Path difference induced by the presence of the object:

FIG. 45 shows the principle for calculating this path difference. Wehave:

Δ=l_(o)n_(o)−l_(v)n_(v) in which n_(v) designates the nominal index ofthe objectives, i.e. the index for which they were designed and whichmust be that of the optical oil used.

Δ=Ln_(o) cos β−Ln_(v) cos α$\Delta = {L\left( {{n_{o}\sqrt{1 - \left( {\frac{n_{v}}{n_{o}}\sin \quad \alpha} \right)^{2}}} - {n_{v}\sqrt{1 - {\sin^{2}\alpha}}}} \right)}$

 hence:$\Delta = {L\left( {{n_{o}\sqrt{1 - {\left( \frac{n_{v}}{n_{o}} \right)^{2}\frac{i^{2} + j^{2}}{K_{p}^{2}}}}} - {n_{v}\sqrt{1 - \frac{i^{2} + j^{2}}{K_{p}^{2}}}}} \right)}$

Path difference induced by the displacement of the illuminated point:

The parameter accessible for measurement is the path difference betweenthe inverted beam coming from FRG and the reference beam FRD. If thesetwo beams coincide and if the index of the medium separating theobjectives is the nominal index of the objectives, the path differenceis zero. However, if the virtual origin of the reference beam FRD hasthe coordinates (x,y,z) in relation to the virtual origin of thereference beam FRG, materialized by the point of focus of its invertedbeam, then this path difference is calculated as follows:

The vector v=(x,y,z) and the vector u defined above are used. FIG. 46shows the geometrical calculation principle. The path difference inducedis:$\Delta = {{u \cdot v} = {\left( {\frac{i}{K_{p}},\frac{j}{K_{p}},{\frac{1}{K_{p}}\sqrt{K_{p}^{2} - i^{2} - j^{2}}}} \right) \cdot \left( {x,y,z} \right)}}$$\Delta = {{x\frac{i}{K_{p}}} + {y\frac{j}{K_{p}}} + {z\frac{1}{K_{p}}\sqrt{K_{p}^{2} - i^{2} - j^{2}}}}$

Total path difference:

This is the sum of the path difference due to the presence of the objectand that due to the non-coincidence of the source points.$\Delta = {{n_{v}\left( {{x\quad \frac{i}{K_{p}}} + {y\quad \frac{j}{K_{p}}} + {z\quad \frac{1}{K_{p}}\sqrt{K_{p}^{2} - i^{2} - j^{2}}}} \right)} + {L\left( {{n_{o}\sqrt{1 - {\left( \frac{n_{v}}{n_{o}} \right)^{2}\frac{i^{2} + j^{2}}{K_{p}^{2}}}}} - {n_{v}\sqrt{1 - \frac{i^{2} + j^{2}}{K_{p}^{2}}}}} \right)}}$

In the absence of a diaphragm, the wave measured at the point of thesensor with coordinates (i,j) is equal to:$^{\overset{\sim}{j}2\pi \frac{\Delta}{\lambda}} = {\exp \left\{ {\overset{\sim}{j}{\frac{2\pi}{\lambda}\left\lbrack {{n_{v}\left( {{x\quad \frac{i}{K_{p}}} + {y\quad \frac{j}{K_{p}}} + {z\quad \frac{1}{K_{p}}\sqrt{K_{p}^{2} - i^{2} - j^{2}}}} \right)} + {L\left( {{n_{o}\sqrt{1 - {\left( \frac{n_{v}}{n_{o}} \right)^{2}\frac{i^{2} + j^{2}}{K_{p}^{2}}}}} - {n_{v}\sqrt{1 - \frac{i^{2} + j^{2}}{K_{p}^{2}}}}} \right)}} \right\rbrack}} \right\}}$

7.8. Maximization program

7.8.1. Principle

In the “ideal” case in which the sample is simply a plate of thickness Land refractive index n_(o), a simple method may be used to determine thevalues of x,y,z,L, n_(o). For this adjustment, the illuminating beamsFED and FEG are eliminated and the reference beams FRD and FRG areintroduced using the mirrors (2282) and (2247) to send symmetrical beamsto the objectives. The beam FRG reflected by (2282) and directed towardsthe objective (2219) is centered on a central point of the diaphragm(2224). It is focussed by the objective (2219) at a point of the object(2218). It then passes through the objective (2217) and reaches the CCD(2239) on which it is superimposed on the reference beam FRD reflectedby the semi-transparent mirror (2245) in the direction of the CCD(2239). The phase shifter (2251) is used to generate a phase shiftbetween the two beams, thus allowing the complex value measurement ofthe beam coming from FRG and received on (2239), using the simpleimaging procedure described in 7.5. The wave thus received is stored inan array F_(rec) of dimensions N_(pix)×N_(pix).

In practice, the acquisition method will often differ from the idealcase but will always allow the acquisition of an array F_(rec) ofdimensions N_(pix)×N_(pix) equivalent to the one that would be obtainedin the ideal case.

Owing to the phase differences induced by the sample, the value of thewave at a point in the absence of a diaphragm, and in a system ofcoordinates centered on the optical center, is given by the formulaestablished in 7.7.:$^{\overset{\sim}{j}2\pi \frac{\Delta}{\lambda}} = {\exp \left\{ {\overset{\sim}{j}{\frac{2\pi}{\lambda}\left\lbrack {{n_{v}\left( {{x\quad \frac{i}{K_{0}}} + {y\quad \frac{j}{K_{0}}} + {z\quad \sqrt{1 - \frac{i^{2} + j^{2}}{K_{0}^{2}}}}} \right)} + {L\left( {{n_{o}\sqrt{1 - {\left( \frac{n_{v}}{n_{o}} \right)^{2}\frac{i^{2} + j^{2}}{K_{0}^{2}}}}} - {n_{v}\sqrt{1 - \frac{i^{2} + j^{2}}{K_{0}^{2}}}}} \right)}} \right\rbrack}} \right\}}$

The presence of the diaphragm moreover has a filtering effect which mayalso be simulated.

The purpose of the program used is to calculate x,y,z,L, n_(o) byminimizing the difference between the wave thus simulated and the waveactually received.

This difference may be characterized by the standard deviationσ(x,y,z,L,n_(o)) between the simulated wave and the received wave. Asthe received wave is stored in an array F_(rec) of dimensionsN_(pix)×N_(pix), this standard deviation may be calculated as follows,in six stages, for a given value of the quintuplet (x,y,z,L,n_(o)):

Stage 1—The following frequency representation, of dimensionsN_(c)×N_(c) with, for example, N_(c)=4096, is calculated:${F_{c}\left\lbrack {i,j} \right\rbrack} = {\exp \left\{ {\overset{\sim}{j}{\frac{2\pi}{\lambda}\left\lbrack {{n_{v}\left( {{x\quad \frac{ic}{R_{c}}} + {y\quad \frac{jc}{R_{c}}} + {z\quad \sqrt{1 - \frac{{ic}^{2} + {jc}^{2}}{R_{c}^{2}}}}} \right)} + {L\left( {{n_{o}\sqrt{1 - {\left( \frac{n_{v}}{n_{o}} \right)^{2}\frac{{ic}^{2} + {jc}^{2}}{R_{c}^{2}}}}} - {n_{v}\sqrt{1 - \frac{{ic}^{2} + {jc}^{2}}{R_{c}^{2}}}}} \right)}} \right\rbrack}} \right\}}$

 where $R_{c} = {\frac{N_{c}}{N_{pix}}K_{0}}$

 and where ic and jc are the indices centered:${{ic} = {i - \frac{N_{c}}{2}}},{{jc} = {j - \frac{N_{c}}{2}}}$

Stage 2—The inverse Fourier transform of F_(c) is calculated.

Stage 3—F_(c) now being in spatial representation, the presence of thediaphragm constitutes a simple limitation in this domain. The programthus calculates the table F_(d) of dimensions N_(pix)×N_(pix) byinitializing it to 0 and then performing:${F_{d}\left\lbrack {i,j} \right\rbrack} = {F_{c}\left\lbrack {{i - \frac{N_{pix}}{2} + \frac{N_{c}}{2}},{j - \frac{N_{pix}}{2} + \frac{N_{c}}{2}}} \right\rbrack}$

 for all the pairs (i,j) such that${\left( {i - \frac{N_{pix}}{2}} \right)^{2} + \left( {j - \frac{N_{pix}}{2}} \right)^{2}} \leq \left( \frac{N_{pix}}{2} \right)^{2}$

Stage 4—The program performs a Fourier transformation of the table F_(d)

Stage 5—The program calculates the value${rapport} = \frac{\sum\limits_{{({i,j})} \in E}{{F_{rec}\left\lbrack {i,j} \right\rbrack}\overset{\_}{F_{d}\left\lbrack {i,j} \right\rbrack}}}{\sum\limits_{{({i,j})} \in E}{{F_{d}\left\lbrack {i,j} \right\rbrack}}^{2}}$

 in which E is a set defined in order to avoid excessively attenuatedvalues, allowing phase and intensity correction on the table F_(d)before comparing it to F_(rec).

Stage 6—The program then calculates the standard deviation${\sigma^{2}\left( {x,y,z,L,n_{o}} \right)} = {\sum\limits_{i,j}{{{F_{rec}\left\lbrack {i,j} \right\rbrack} - {{rapport} \cdot {F_{d}\left\lbrack {i,j} \right\rbrack}}}}^{2}}$

The calculation program of x,y,z,L, n_(o) determines the value ofx,y,z,L, n_(o) which minimizes the standard deviation thus calculated.Any program for minimizing the value σ²(x,y,z,L,n_(o)) may be used in anequivalent manner. The algorithm described in 7.8.2. constitutes anexample of such a program but may be replaced by any equivalentminimization program.

To simplify the calculations and facilitate convergence, the minimizingof σ²(x,y,z,L,n_(o)) is replaced in the algorithm described in 7.8.2. bythe maximizing of a characteristic value which varies according to amode variable which increases as the algorithm converges towards thesolution. Three values of mode are used:

Mode=1: The image actually received is phase-corrected by multiplicationby e^(−{tilde over (j)}2πΔ/λ). An inverse Fourier transform allows aspatial representation to be obtained. The value chosen is the maximumof the modulus over the entire spatial representation.

Mode=2: Same as mode=1, but the central part of the spatialrepresentation is over-sampled and the value chosen is the value of themodulus at the barycenter of the points of the spatial representation.

Mode=3: The wave to be received on the sensor for the considered valuesof x,y,z,L, n_(o) is calculated taking into account the filtering by thediaphragm. The value chosen is the opposite of the standard deviationbetween the wave thus simulated and the wave actually received, namely−σ²(x,y,z,L,n_(o))

7.8.2. Algorithm

The algorithm of this program is described in FIGS. 47 to 50 and FIG.60.

FIG. 47 describes the highest level of the program. This level consistsof a double variation loop for the index n_(o). The program calculatesnopixels values of maxb, the highest value reached by the characteristicvalue for the index n_(o), between nomin and nomax. This corresponds tothe internal loop (3201). The program determines the new value ofnocentre, which must correspond to the maximum value of maxb. It thenbegins a new iteration of the type (3201) where the values of n_(o) arecentered around the new value of nocentre and where the widthnolarg=nomax−nomin of the search interval has been divided by 2. Thisconstitutes the external loop (3202) which is reiterated until the widthof the search interval corresponds to the precision sought on the indexn_(o). This method obviates the need to test too many values of n_(o) toreach a result of a given precision.

The program must have the following input values:

minimum and maximum values of each value sought based on availableinformation. nomin_ini, nomax_ini (index n_(o)), Lmin,Lmax (width L),zmin,zmax (depth z). The program does not require a maximum and minimumvalue of the coordinates x and y.

operating parameters, for example nopixels=5 and pixels=50

image obtained, for example in the manner described in 7.8.1., in theform of an array of complex numbers F_(rec)[i,j] of dimensionsN_(pix)×N_(pix).

The main steps in the program are:

(3203): the current value of n_(o) is calculated.

(3204): this procedure calculates the maximum value maxb reached by thecharacteristic value for the current index n_(o), as well as thecorresponding values of x,y,z,L. It is illustrated in detail in FIG. 48.

(3205): When the value maxb corresponding to the current iteration isgreater than max_no, the current values of x,y,z,L, (calculated usingthe procedure (3204)), and n_(o) are stored and constitute the currentapproximation of the sought result.

(3206): The width is compared with a certain limit to determine theconvergence condition. It is possible to have, for example, lim=0.05λ/Lnopixels

(3207): The program ends. The values x_(f),y_(f),z_(f),L_(f), nocentrecorrespond to the best approximation of the real values of x,y,z,L,n_(o). They are displayed and stored for subsequent re-use.

For each index n_(o), the procedure (3204) calculates a maximum value ofthe characteristic value and the associated values of x,y,z,L. However,a change of variables is made and the variables really used in theprocedure are x,y,u,v where:

u=cL+z

v=L−cz

$c = {\frac{n_{v}}{n_{o}} - 1}$

where n_(v) is the nominal refractive index of the objectives.

The procedure consists in varying u and v and, for each pair (u,v), incalculating x,y, and the max value of the characteristic value.

The pair (u,v) varies initially over a discrete set of points of sizeupixels x vpixels, u and v varying respectively over the width intervalsularg and vlarg centered around the points ucentre_ini and vcentre_ini.The program determines the new value of (ucentre,vcentre) whichcorresponds to the pair (u, v) for which the max value is the highest.It also determines values xbar and ybar which correspond to thedeviation between the values of x,y used for the calculation and therefined value of x,y after the calculation. These operations constitutethe procedure (3305) shown in detail in FIG. 49.

When a new value of (ucentre,vcentre) has been obtained, the programdecreases the width ularg and vlarg of the search intervals, as well astheir centers ucentre_ini, vcentre_ini and the values of x and y, thencalculates the value Δdif. If the obtained value is less than a limitset as convergence criterion, the program calculates z and L byinverting the change of variables, thus terminating the procedure(3204). Otherwise, it modifies if necessary the choice of characteristicvalue (mode modification), then repeats the procedure (3305). Thisconstitutes the loop (3301). These operations constitute the procedure(3204) described in FIG. 48.

FIG. 48 describes the procedure (3204). Its basic steps are:

(3302): Variation intervals of u and v are determined.

(3303): This procedure is designed to determine upixels and vpixels inan optimum manner. Its algorithm is described in FIG. 60.

(3304): The deviation of (phase/2π) caused by the crossing of the plateand the displacement along z of the point of impact for a beam having adirection parallel to z is: dif=1/λ((n_(o)−n_(v))L+n_(v)z). It isconsidered that the algorithm has converged when this value is knownwith sufficient accuracy. The uncertainty on this value is equal to:${\Delta \quad {dif}} = {\frac{1}{\lambda}{\frac{1}{1 + c^{2}}\left\lbrack {{\left( {{n_{o}c} - {n_{v}c} + n_{v}} \right)\Delta \quad u} + {\left( {{- n_{o}} + n_{v} + {n_{v}c}} \right)\Delta \quad v}} \right\rbrack}}$

 where${\Delta \quad u} = {{\frac{ularg}{upixels}\quad {and}\quad \Delta \quad v} = \frac{vlarg}{vpixels}}$

 The program modifies mode and determines the end of convergenceaccording to the value obtained for Δdif. It is possible for example tohave lim1=2, lim2=0.25,lim3=0.01

(3305): This procedure calculates the characteristic value for a set ofpairs (u,v) and determines the pair ucentre, vcentre corresponding tothe highest characteristic value, the value maxb of this characteristicvalue, and the values xbar,ybar which represent the deviation betweenthe new approximation of x,y and the current values of x,y. It is shownin detail in FIG. 49.

(3306): Modification of x,y,ularg,vlarg,ucentre_ini,vcentre_ini Theprogram carries out the following modifications:

x=x+xbar

y=y+ybar

if upixels ≦4 then ucentre_ini, ularg are not modified.

if upixels ≧5 then: ucentre_ini=ucentre and${ularg} = {\frac{4}{upixels}{ularg}}$

if vpixels ≦4 then: vcentre_ini, vlarg are not modified.

if vpixels ≧5 then: vcentre_ini=vcentre and${vlarg} = {\frac{4}{vpixels}{vlarg}}$

FIG. 49 describes the procedure (3305). Its basic steps are:

(3401): This procedure calculates the max characteristic value for thecurrent values of u,v, n_(o). In Mode 1 and 2 it also calculates xbarand ybar which represent the deviation between the new approximation ofx,y and the input value of the procedure.

(3402): In the case of Mode 3, the procedure (3401) has not calculatedxbar and ybar. It is restarted in Mode 2 to perform this calculation.

FIG. 50 describes the procedure (3401). Its basic steps are:

(3501): Calculation of corrected frequency representation. From thearray F_(rec)[i,j] the program calculates a corrected representationusing:${a\left\lbrack {i,j} \right\rbrack} = {\frac{1}{\lambda}\left\{ {{\frac{c}{1 + c^{2}}{n_{o}\left( {\sqrt{1 - {\left( \frac{n_{v}}{n_{o}} \right)^{2}\frac{{ic}^{2} + {jc}^{2}}{K_{0}^{2}}}} - 1} \right)}} + {\frac{1 - c}{1 + c^{2}}{n_{v}\left( {\sqrt{1 - \frac{{ic}^{2} + {jc}^{2}}{K_{0}^{2}}} - 1} \right)}}} \right\}}$${b\left\lbrack {i,j} \right\rbrack} = {\frac{1}{\lambda}\left\{ {{\frac{1}{1 + c^{2}}{n_{o}\left( {\sqrt{1 - {\left( \frac{n_{v}}{n_{o}} \right)^{2}\frac{{ic}^{2} + {jc}^{2}}{K_{0}^{2}}}} - 1} \right)}} - {\frac{1 + c}{1 + c^{2}}{n_{v}\left( {\sqrt{1 - \frac{{ic}^{2} + {jc}^{2}}{K_{0}^{2}}} - 1} \right)}}} \right\}}$

where ic and jc are centered indices:${{ic} = {i - \frac{N_{pix}}{2}}},{{jc} = {j - \frac{N_{pix}}{2}}}$

We then verify, to with a constant bias independent of i and j:$\frac{\Delta}{\lambda} = {{\frac{n_{v}}{\lambda}\left( {{x\quad \frac{ic}{K_{0}}} + {y\quad \frac{jc}{K_{0}}}} \right)} + {{a\left\lbrack {i,j} \right\rbrack}u} + {{b\left\lbrack {i,j} \right\rbrack}v}}$

The program generates the corrected frequency representation as follows:${F_{cor}\left\lbrack {i,j} \right\rbrack} = {{F_{rec}\left\lbrack {i,j} \right\rbrack}{\gamma \left( {{a\left\lbrack {i,j} \right\rbrack}\Delta \quad u} \right)}{\gamma \left( {{b\left\lbrack {i,j} \right\rbrack}\Delta \quad v} \right)}\exp \left\{ {{- \overset{\sim}{j}}2{\pi\left\lbrack {{\frac{n_{v}}{\lambda}\left( {{x\quad \frac{i - \frac{N_{pix}}{2}}{K_{0}}} + {y\frac{j - \frac{N_{pix}}{2}}{K_{0}}}} \right)} + {{a\left\lbrack {i,j} \right\rbrack}u} + {{b\left\lbrack {i,j} \right\rbrack}v}} \right\rbrack}} \right\}}$

where γ(x)=0 when |x|≧1/2 and γ(x)=1 when |x|<1/2, and${{\Delta \quad u} = \frac{ularg}{upixels}},{{\Delta \quad v} = \frac{vlarg}{vpixels}}$

a[i,j] and b[i,j] represent respectively the frequencies along u and vof the corrected function obtained. We verify that these frequenciesboth have a constant sign. Multiplication by the γ functions makes itpossible to cancel the elements for which these frequencies are too highand thus avoid aliasing which would prevent the convergence of thealgorithm.

(3502): An inverse Fourier transform of array F_(cor) is performed.

(3503): max is the maximum value of the module in the array F_(cor)[i,j]

 We denote as imax and jmax the coordinates of the pixel in which themaximum is reached. We then have:${xbar} = {\frac{K_{0}\lambda}{N_{v}N_{pix}}\left( {{imax} - \frac{N_{pix}}{2}} \right)}$${ybar} = {\frac{K_{0}\lambda}{N_{v}N_{pix}}\left( {{jmax} - \frac{N_{pix}}{2}} \right)}$

(3504): The central part of the array F_(cor)[i,j] is extracted. We thuscreate a array F_(a)[i,j] of dimensions N_(a)×N_(a) with for exampleN_(a)=16:${F_{a}\left\lbrack {i,j} \right\rbrack} = {F_{cor}\left\lbrack {{i - \frac{N_{a}}{2} + \frac{N_{pix}}{2}},{j - \frac{N_{a}}{2} + \frac{N_{pix}}{2}}} \right\rbrack}$

(3505): A direct Fourier transform is performed on the array F_(a)

(3506): The array F_(a) is completed by zeros and we obtain an arrayF_(b) of dimensions N_(b)×N_(b) with for example N_(b)=512.${F_{b}\left\lbrack {i,j} \right\rbrack} = {F_{a}\left\lbrack {{i - \frac{N_{b}}{2} + \frac{N_{a}}{2}},{j - \frac{N_{b}}{2} + \frac{N_{a}}{2}}} \right\rbrack}$

when${N_{a} - 1} \geq {i - \frac{N_{b}}{2} + \frac{N_{a}}{2}} \geq {{0\quad {and}\quad N_{a}} - 1} \geq {j - \frac{N_{b}}{2} + \frac{N_{a}}{2}} \geq 0$

and F_(b)[i,j]=0 otherwise.

(3507): The Inverse Fourier transform of array F_(b) is performed.

(3508): xbar,ybar,max are calculated by the formulas:${imax} = \frac{\sum\limits_{i,j}{{{F_{b}\left\lbrack {i,j} \right\rbrack}}^{2}i}}{\sum\limits_{i,j}{{F_{b}\left\lbrack {i,j} \right\rbrack}}^{2}}$${jmax} = \frac{\sum\limits_{i,j}{{{F_{b}\left\lbrack {i,j} \right\rbrack}}^{2}j}}{\sum\limits_{i,j}{{F_{b}\left\lbrack {i,j} \right\rbrack}}^{2}}$

 max=|F _(b) [imax, jmax]|

${xbar} = {\frac{K_{0}\lambda}{n_{v}N_{pix}}\frac{N_{a}}{N_{b}}\left( {{imax} - \frac{N_{b}}{2}} \right)}$${ybar} = {\frac{K_{0}\lambda}{n_{v}N_{pix}}\frac{N_{a}}{N_{b}}\left( {{jmax} - \frac{N_{b}}{2}} \right)}$

(3509): The following frequency representation, of dimensionsN_(c)×N_(c) with for example N_(c)=4096, is calculated:${F_{c}\left\lbrack {i,j} \right\rbrack} = {\exp \left\{ {\overset{\sim}{j}{\frac{2\pi}{\lambda}\left\lbrack {{n_{v}\left( {{x\frac{ic}{R_{c}}} + {y\frac{jc}{R_{c}}} + {z\sqrt{1 - \frac{{ic}^{2} + {jc}^{2}}{R_{c}^{2}}}}} \right)} + {L\left( {{n_{o}\sqrt{1 - {\left( \frac{n_{v}}{n_{o}} \right)^{2}\frac{{ic}^{2} + {jc}^{2}}{R_{c}^{2}}}}} - {n_{v}\sqrt{1 - \frac{{ic}^{2} + {jc}^{2}}{R_{c}^{2}}}}} \right\}}} \right\rbrack}} \right\}}$

 where $R_{c} = {\frac{N_{c}}{N_{pix}}K_{0}}$

 and where ic and jc are centered indices:${{ic} = {i - \frac{N_{c}}{2}}},{{jc} = {j - \frac{N_{c}}{2}}}$

(3510): The inverse Fourier transform of F_(c) is calculated.

(3511): F_(c) now being in spatial representation, the presence of thediaphragm constitutes a simple limitation in this domain. The programthus calculates the array F_(d) of dimensions N_(pix)×N_(pix) byinitializing to 0 and then carrying out:${F_{d}\left\lbrack {i,j} \right\rbrack} = {F_{c}\left\lbrack {{i - \frac{N_{pix}}{2} + \frac{N_{c}}{2}},{j - \frac{N_{pix}}{2} + \frac{N_{c}}{2}}} \right\rbrack}$

 for all the pairs (i,j) such that${\left( {i - \frac{N_{pix}}{2}} \right)^{2} + \left( {j - \frac{N_{pix}}{2}} \right)^{2}} \leq \left( \frac{N_{pix}}{2} \right)^{2}$

(3512): The program performs a Fourier transform of array F_(d)

(3513): The program calculates the value${rapport} = \frac{\sum\limits_{{({i,j})} \in E}{{F_{rec}\left\lbrack {i,j} \right\rbrack}\overset{\_}{F_{d}\left\lbrack {i,j} \right\rbrack}}}{\sum\limits_{{({i,j})} \in E}{{F_{d}\left\lbrack {i,j} \right\rbrack}}^{2}}$

 where E is the set of pairs (i, j) complying with:${{{F_{rec}\left\lbrack {i,j} \right\rbrack}\overset{\_}{F_{d}\left\lbrack {i,j} \right\rbrack}} \geq {{Coef} \cdot {\max\limits_{\underset{0 \leq b \leq {N_{pix} - 1}}{0 \leq a \leq {N_{pix} - 1}}}{{{F_{rec}\left\lbrack {a,b} \right\rbrack}\overset{\_}{F_{d}\left\lbrack {a,b} \right\rbrack}}}}}},$

 with for example Coef=0,5. The justification for this formula can befound in 7.17.1.2.

(3514): The program then calculates the max characteristic value:$\max = {- {\sum\limits_{i,j}{{{F_{rec}\left\lbrack {i,j} \right\rbrack} - {{rapport} \cdot {F_{d}\left\lbrack {i,j} \right\rbrack}}}}^{2}}}$

FIG. 60 describes the procedure (3303). This procedure seeks todetermine upixels and vpixels according to the principles set forthbelow:

It is possible to express the value Δ/λ calculated in 7.7., with aconstant bias, in the following form:$\frac{\Delta}{\lambda} = {{\frac{n_{v}}{\lambda}\left( {{x\frac{i}{K_{0}}} + {y\frac{j}{K_{0}}}} \right)} + {{a(s)}u} + {{b(s)}v}}$

where:${a(s)} = {\frac{1}{\lambda}\left\{ {{\frac{c}{1 + c^{2}}{n_{o}\left( {\sqrt{1 - {\left( \frac{n_{v}}{n_{o}} \right)^{2}s^{2}}} - 1} \right)}} + {\frac{1 - c}{1 + c^{2}}{n_{v}\left( {\sqrt{1 - s^{2}} - 1} \right)}}} \right\}}$${b(s)} = {\frac{1}{\lambda}\left\{ {{\frac{c}{1 + c^{2}}{n_{o}\left( {\sqrt{1 - {\left( \frac{n_{v}}{n_{o}} \right)^{2}s^{2}}} - 1} \right)}} - {\frac{1 - c}{1 + c^{2}}{n_{v}\left( {\sqrt{1 - s^{2}} - 1} \right)}}} \right\}}$

${s = \sqrt{\frac{i^{2} + j^{2}}{K_{0}^{2}}}},$

i and j being centered indices.

The method for generating the characteristic value, in the case ofmode=1 or mode=2, consists in multiplying the frequency representationF_(rec) by the phase correction factor e^(−j2πΔ/λ) and then performing aFourier transform. The representation thus obtained is in the spatialdomain. The characteristic value is roughly the value of thisrepresentation at the origin. It is thus obtained as a sum of elementsof the form e^(−j2πΔ/λ). This characteristic value may be considered tobe a ftmctionfonc(u,v) of u and v. If we fix the value of v, it becomessimply a function of u. If we limit the representation F_(rec) to a discof radius (in pixels) K₀s_(o), this function of u has as its maximumfrequency a(s_(o)). a(s_(o)) is thus the maximum frequency of fonc alongu. Similarly, b(s_(o)) is the maximum frequency of fonc along v. upixelsand vpixels must be determined so that, for a given value of s₀, as highas possible, the sampling intervals along u and v are sufficientlyprecise to prevent aliasing. This condition is written:${{{a\left( s_{o} \right)}}\frac{ularg}{upixels}} = {{{{b\left( s_{o} \right)}}\frac{vlarg}{vpixels}} = \frac{1}{2}}$

Further, the total number of pixels is limited to a pixels value, whichis written:

upixels.vpixels=pixels

Combining these equations we obtain the equation E1:${{{a\left( s_{o} \right)}{b\left( s_{o} \right)}}} = \frac{pixels}{4 \cdot {ularg} \cdot {vlarg}}$

Two cases may then arise:

first case:${{{a\left( \frac{ouv}{n_{v}} \right)}{b\left( \frac{ouv}{n_{v}} \right)}}} \geq \frac{pixels}{4 \cdot {ularg} \cdot {vlarg}}$

The equation E1 has a solution s_(o) that the program can determine bydichotomy. The values of upixels and vpixels are then determined by:

upixels=2|a(s _(o))|ularg

vpixels=2|b(s _(o))|vlarg

Second case:${{{a\left( \frac{ouv}{n_{v}} \right)}{b\left( \frac{ouv}{n_{v}} \right)}}} < \frac{pixels}{4 \cdot {ularg} \cdot {vlarg}}$

The equation E1 has no solution. The values of upixels and vpixels arethen determined to be proportional to those obtained for the value$s_{o} = {\frac{ouv}{n_{v}}\quad {i.e}\text{:}}$

${upixels} = {\alpha {{a\left( \frac{ouv}{n_{v}} \right)}}{ularg}}$${vpixels} = {\alpha {{b\left( \frac{ouv}{n_{v}} \right)}}{vlarg}}$

Also, the condition upixels.vpixels=pixels must always be true. Thesolution of these equations gives:$\alpha^{2} = \frac{pixels}{{{{a\left( \frac{ouv}{n_{v}} \right)}{b\left( \frac{ouv}{n_{v}} \right)}}}{{ularg} \cdot {vlarg}}}$

and we thus obtain finally:${upixels} = {\sqrt{\frac{{{a\left( \frac{ouv}{n_{v}} \right)}}{ularg}}{{{b\left( \frac{ouv}{n_{v}} \right)}}{vlarg}}}\sqrt{pixels}}$${vpixels} = {\sqrt{\frac{{{b\left( \frac{ouv}{n_{v}} \right)}}{vlarg}}{{{a\left( \frac{ouv}{n_{v}} \right)}}{ularg}}}\sqrt{pixels}}$

In both cases, the values thus determined are real values which may beless than 1 or higher than pixels. A last step thus consists intranslating them into integer numbers within the interval [1,pixels].

The resulting algorithm, which allows the determination of upixels andvpixels, is represented in FIG. 60. The following steps must be examinedin detail:

(4201): s_(o) is the solution of the equation${{{a\left( s_{o} \right)}{b\left( s_{o} \right)}}} = {\frac{pixels}{4 \cdot {ularg} \cdot {vlarg}}.}$

 The program solves this equation by dichotomy between 0 and$\frac{ouv}{n_{v}}.$

(4202): The program takes for upixels and vpixels the integer numbernearest to the real value obtained, then limits these values as follows:

if upixels<1 the program performs upixels=1

if vpixels<1 the program performs vpixels=1

if upixels>pixels the program performs upixels=pixels

if vpixels>pixels the program performs vpixels=pixels

7.9. Obtaining characteristics of parallel beams

Each parallel beam generated by the illuminating beam control system hasan independent phase. The purpose of this procedure is to determine thephases and the coordinates of these parallel beams. It is broken downinto two parts:

7.9.1. Adjustment of objectives and obtaining the relative position ofthe points of focus

The purpose of the present step is to determine the relative position ofthe origins of the reference beams FRD and FRG. This may be achieved byadjusting the position of the objectives so that the image produced bythe beam FRGI on the sensor (2239) is a perfect point image.

For this purpose, only the beams FRGI and FRD are used. The mirror(2282) is used to obtain a centered wave with two propagationdirections. The mirrors (2243) (2232) (2247) are blocked. The phaseshifter (2251) is used to modify the phase of this wave. The spacebetween the two objectives is occupied by optical oil with the nominalindex of the objectives. The wave measured on the right side of theobjectives on the sensor (2239) indexed by p=0 is the equivalent of thereference wave used on the other side. Its punctuality is evaluated bythe procedure described in 7.3.3.1., using a Fourier transform in saidprocedure. The position of the objectives is set to obtain a pointcentered in the middle of the image.

If this objective position adjustment is carried out with the greatestcare and with sufficiently precise positioners, of sub-micrometricprecision, for example microscope focussing devices of sufficientquality, or positioners with piezoelectric control, then the origin ofthe images obtained on each side of the microscope match.

This however requires extremely precise positioning of the objectives.The adjustment procedure described in 7.3.3.2. however makes it possibleto obtain this adjustment quality using positioners of average precisionfor the objectives. In fact, fine adjustments are carried out in thisprocedure by moving the cameras and not the objectives. If theadjustment procedure described in 7.3.3.2. is carried out with care, theorigins of the frequency representations finally obtained will match.Their relative coordinates are thus (x,y,z)=(0,0,0). It is thenpreferable not to modify the positions obtained for the objectives.

However, the adjustments obtained by procedure 7.3.3.2. or by a newadjustment of the position of the objectives are in general not perfect.It is possible to use (x,y,z)=(0,0,0), but a better superposition of theimages coming from each sensor will be obtained if a suitable algorithmis used to calculate a more precise value of these parameters.

The wave measured on the sensor (2239), obtained according to theprocedure described in 7.5. with p=0, is then stored in an arrayF_(rec)[i,j]. The coordinates x,y,z of the origin of the reference wavecoming from FRRD in relation to the origin of the reference wave comingfrom FRG are then determined from F_(rec)[i,j] using the programdescribed in 7.8. However, in order to take into account the absence ofa sample, the procedure (3303) is replaced by the following twoassignments:

upixels=pixels

vpixels=1

Further, the variables used are:

nomax_ini=nomin_ini=n_(v), n_(v) being the nominal index of theobjectives.

Lmin=Lmax=0

nopixels=1

pixels=20

zmin=−20λ and zmax=20λ (for example)

7.9.2. Obtaining complex values and coordinates of the illuminatingbeams

The purpose of this procedure is to determine, for each parallel beamdefined by the indices p,i,j of the control word COM[p,i,j] of the beamdeflector:

the coordinates of the point of direct impact of the illuminating beamon the sensor p. These coordinates will be stored in the arraysIa[p,0,i,j], Ja[p,0,i,j].

the coordinates of the point of direct impact of the illuminating beamon the sensor {overscore (p)}. These coordinates will be stored in thearrays Ia[p,1,i,j], Ja[p,1,i,j].

the complex value of the corresponding illuminating beam, its phasebeing measured at the origin of the reference wave used on the sensor{overscore (p)}. This phase measurement convention in fact ensures thatthe complex value thus obtained is independent of the position of theobjectives, to within a global phase factor affecting all the beamscharacterized by the same index p. This complex value will be stored inan array Ra[p,i,j]. As the direct measurement of the beams givescoordinates relative to the origin of the reference wave used on thesensor p, this value must be corrected according to the position valuesdetermined en 7.9.1.

During this procedure, the reference beams FRD and FRG are used, as wellas an illuminating beam defined by a variable control word COM[p,i,j] ofthe beam deflector. The mirrors (2243) and (2232) are used to create anopposite indicator beam of the illuminating beam which will strike thesensor opposite the sensor normally illuminated by this beam. Themirrors (2247) and (2282) are blocked. The procedure described in 7.5.is used to obtain, on each sensor, two-dimensional images in complexnumbers. The phase shifts are obtained by means of the element (2205).

A program makes it possible to obtain these parameters in the form ofarrays Ia[p,q,i,j], Ja[p,q,i,j], Ra[p,i,j]. Using an oversamplingmethod, it determines Ia[p,q,i,j], Ja[p,q,i,j] with sub-pixel precision.These arrays are thus arrays of real numbers.

The coordinate system used on the sensor p is that determined in Step 14of the adjustment cycle in 7.3.3.2., defined by the direction vectors({right arrow over (i)}_(p),{right arrow over (j)}_(p)). This coordinatesystem convention will be maintained hereinafter.

The program is made up of three loops covered successively.

Loop 1: This is a loop on the index p, which takes on the values 0 and 1successively. For each of these indices the program uses the controlword${{COM}\left\lbrack {p,\frac{N_{pix}}{2},\frac{N_{pix}}{2}} \right\rbrack}.$

 It then determines the integer coordinates imax0 _(p), jmax0 _(p) ofthe maximum of the resulting image on the sensor p.

Loop 2: This is a loop on the set of triplets p₀,i₀,j₀. For each ofthese triplets, the program generates the values Ia[p₀,q,i₀,j₀]Ja[p₀,q,i₀,j₀] Ra[p₀,i₀,j₀]. With each iteration, corresponding to agiven triplet p₀,i₀,j₀ the program completes the following 8 steps:

Step 1: The control word${COM}\left\lbrack {p_{0},\frac{N_{pix}}{2},\frac{N_{pix}}{2}} \right\rbrack$

is used and the resulting images on each sensor are stored in two arraysof complex numbers M0 _(q)[i,j] where q=0 for the image obtained onsensor p₀ and q=1 for the image obtained on the opposite sensor.

Step 2: The control word COM[p₀,i₀,j₀] is used and the resulting imagesare stored in the arrays M1 _(q)[i,j]

Step 3: The program determines the coordinates imax_(q), jmax_(q) of thepoint corresponding to the modulus maximum value of the array M1_(q)[i,j] for each value of q.

Step 4: The program extracts an image of size N_(a)×N_(a), with forexample N_(a)=16, around the point of coordinates imax_(q) jmaX_(q):${{M2}_{q}\left\lbrack {i,j} \right\rbrack} = {{M1}_{q}\left\lbrack {{i - \frac{N_{a}}{2} + {imax}_{q}},{j - \frac{N_{a}}{2} + {jmax}_{q}}} \right\rbrack}$

 when:${N_{pix} - 1} \geq {i - \frac{N_{a}}{2} + {imax}_{q}} \geq {{0\quad {and}\quad N_{pix}} - 1} \geq {j - \frac{N_{a}}{2} + {jmax}_{q}} \geq 0$

 and M2 _(q)[i,j]=0 otherwise.

Step 5: The program performs a direct Fourier transform of the arrays M2_(q).

Step 6: The program completes the arrays M2 _(q) by zeros, generatingthe arrays M3 _(q) of dimensions N_(b)×N_(b) with for example N_(b)=512.The array M3 _(q) is set to zero and then the program carries out, forall the indices i,j each ranging from 0 to N_(a)−1 and for the twoindices q:${{M3}_{q}\left\lbrack {{i - \frac{N_{a}}{2} + \frac{N_{b}}{2}},{j - \frac{N_{a}}{2} + \frac{N_{b}}{2}}} \right\rbrack} = {{M2}_{q}\left\lbrack {i,j} \right\rbrack}$

Step 7: The program performs the inverse Fourier transform of the arrayM3.

Step 8: The program calculates the coordinates and the complex value atthe barycenter of the array M3.${imax3}_{q} = \frac{\sum\limits_{i,j}{{{{M3}_{q}\left\lbrack {i,j} \right\rbrack}}^{2}i}}{\sum\limits_{i,j}{{{M3}_{q}\left\lbrack {i,j} \right\rbrack}}^{2}}$${jmax3}_{q} = \frac{\sum\limits_{i,j}{{{{M3}_{q}\left\lbrack {i,j} \right\rbrack}}^{2}j}}{\sum\limits_{i,j}{{{M3}_{q}\left\lbrack {i,j} \right\rbrack}}^{2}}$${{Ia}\left\lbrack {p_{0},q,i_{0},j_{0}} \right\rbrack} = {{imax}_{q} + {\frac{N_{a}}{N_{b}}\left( {{imax3}_{q} - \frac{N_{b}}{2}} \right)}}$${{Ja}\left\lbrack {p_{0},q,i_{0},j_{0}} \right\rbrack} = {{jmax}_{q} + {\frac{N_{a}}{N_{b}}\left( {{jmax3}_{q} - \frac{N_{b}}{2}} \right)}}$${{Ra}\left\lbrack {p_{0},i_{0},j_{0}} \right\rbrack} = \frac{{M3}_{0}\left\lbrack {{imax3}_{0},{jmax3}_{0}} \right\rbrack}{{M0}_{0}\left\lbrack {{imax0}_{p_{0}},{jmax0}_{p_{0}}} \right\rbrack}$$\exp \left( {{- \overset{\sim}{j}}\frac{2\quad \pi}{\lambda}{n_{v}\left( {{x\frac{{Ia}\left\lbrack {p_{0},0,i_{0},j_{0}} \right\rbrack}{K_{p_{0}}}} + {y\frac{{Ja}\left\lbrack {p_{0},0,i_{0},j_{0}} \right\rbrack}{K_{p_{0}}}} + {z\sqrt{1 - \frac{{{Ia}\left\lbrack {p_{0},0,i_{0},j_{0}} \right\rbrack}^{2} + {{Ja}\left\lbrack {p_{0},0,i_{0},j_{0}} \right\rbrack}^{2}}{K_{p_{0}}^{2}}}}} \right)}} \right)$

 where x,y,z are the coordinates calculates in 7.9.1.

Loop 3: The program carries out a last operation consisting in cancelingthe values of Ra and in assigning high values to Ja and Ia whenever thepoint of direct impact of the beam is outside the zone limited by theaperture of the objective, resulting in the disappearance of this point.For this purpose, the program again goes through the indices p₀, i₀, j₀,testing each time the condition${\frac{{Ra}\left\lbrack {p_{0},i_{0},j_{0}} \right\rbrack}{{Ra}\left\lbrack {{p_{0},127}{,127}} \right\rbrack}} \leq {\frac{1}{8}.}$

 When this condition is met, the program performs:

Ra[p₀,i₀,j₀]=0

Ia[p₀,0,i₀,j₀]=−1000

Ja[p₀,0,i₀,j₀]=−1000

7.10. Objective position adjustment

The sample to be studied is put in place. During this step, only thebeams FRGI and FRD are used. The mirror (2282) is used to obtain acentered wave with two propagation directions. The mirrors (2243) (2232)(2247) are blocked. The phase shifter (2251) is used to modify the phaseof this wave. Optical oil with the nominal index of the objectives isused. The wave measured on the right side of the objectives on thesensor (2239) indexed by p=0 is the equivalent of the reference waveused on the other side. A program generates two images by the proceduredescribed in 7.3.3.1.: a spatial image obtained with Fouriertransformation, and a frequency image obtained without Fouriertransformation. The program extracts from each of these images theintensity (square of the modulus of the complex numbers constituting theimage obtained in 7.3.3.1.). The program displays the resulting images.

The spatial image must be centered.

On the frequency image, a clear disc must be observed. The adjustmentmust be carried out so that the intensity is the highest possible forthe high frequencies (points far from the center). The observed discmust remain relatively homogeneous.

If a dark ring appears between the outer edge and the central zone, thesample is too thick and it will not be possible to take all thefrequencies into account. The adjustment must then be carried out so asto have a relatively homogeneous disc with as large a radius aspossible. The disc does not reach its maximum size and the highfrequencies cannot be taken into account. The resolution of the image,mainly in depth, is reduced. The only solution to this problem is to usea specially designed objective, described in Paragraph 7.19.

This objective adjustment method is suited to the case in which theindex of the sample differs significantly from the nominal index of theobjectives. In the opposite case, and in particular if one wishes to usea function D_(p) defined in 7.16 equal to 1, the adjustment must becarried out from the spatial image alone and so that this image is acentered point image.

7.11. Determination of x,v,z,L, n₀.

It is necessary to know these parameters in order to be able tocompensate for the spherical aberration and the effects of non-matchingof the origins of the reference waves. These parameters can bedetermined by not carrying out Step 7.10. and by thus leaving theobjectives in the position in which they were at the end of the stepdescribed in 7.9.1. The values x,y,z are then those which weredetermined in 7.9.1. The values of L and n₀ may have been measuredpreviously by a means outside the microscope proper. This method howeverhas the disadvantage of not allowing any movement of the objectives whenputting the sample in place, and of requiring the use of a costlyexternal measurement system. It is thus preferable to determine anew allthese parameters. This second method allows the movement of theobjectives when putting the sample in place and Step 7.10. can thus becarried out.

To apply this second method, it would in principle be possible to usethe theoretical procedure described in 7.8.1. which uses a single imageacquisition, but local variations in the characteristics of the samplenear the point of focus would distort the result. For this reason, aseries of images are taken and, on each image, a measurement is made ofthe phase and intensity variation at the point of direct impact of theilluminating wave. From this series of values, it is possible togenerate an array equivalent to the array F_(rec) used in 7.8., but inwhich local variations are overcome.

7.11.1. Acquisition

The mirrors (2282) (2247) (2243) (2232) are blocked so as to eliminateall the opposite indicator beams, which will no longer be usedhereinbelow. The procedure described in 7.9.2. is restarted, with thefollowing modifications:

The sample is now present.

Loop 1, which consists in determining imax0 _(p), jmax0 _(p), iseliminated. The values previously obtained for imax0 _(p), jmax0 _(p)are used again.

In Loop 2, the indices p₀ and q are set at 0. Steps 1 to 8 are thuscarried out only for the set of indices i₀,j₀ (p₀ being set at 0). Ineach of these steps, only the elements corresponding to the index q=0are acquired or calculated.

Step 8 of Loop 2 is modified as follows:

Ia[p₀,q,i₀,j₀], Ja[p₀,q,i₀,j₀], Ra[p₀,i₀,j₀] are not recalculated andtheir previously obtained values are maintained.

The quantity Rb[p₀,i₀,j₀] is calculated as follows:

if Ra[p₀,i₀,j₀]=0 then Rb[p₀,i₀,j₀]=0

otherwise:${{Rb}\left\lbrack {p_{0},i_{0},j_{0}} \right\rbrack} = {\frac{{M3}_{0}\left\lbrack {{imax3}_{0},{jmax3}_{0}} \right\rbrack}{{M0}_{0}\left\lbrack {{imax0}_{0},{jmax0}_{0}} \right\rbrack}\frac{1}{{Ra}\left\lbrack {p_{0},i_{0},j_{0}} \right\rbrack}}$

This quantity corresponds to the variation of the illuminating beam dueto the presence of the sample and to the movement of the objectives, onthe sensor 0, for the illuminating characterized by the indices i₀,j₀,in relation to a position of the objectives in which the origin of thereference waves used on each objective would match.

Loop 3 is eliminated.

7.11.2. Generation of frequency image

A second program is then started. Its aim is to generate, from theprevious measurements, an array F_(rec) utilizable in the algorithmdescribed in 7.8. As the coordinates Ia[p₀,q,i₀,j₀], Ja[p₀,q,i₀,j₀] ofthe points sampled by the array Rb do not correspond to whole pixels, anoversampling and filtering method is required to generate the arrayF_(rec). This method is not perfect because the interval (for exampleIa[p,q,i,j]−Ia[p,q,i+1,j]) between two adjacent samples of the array Rbcan vary. Nevertheless, for a realization of good quality, thisvariation is small and the oversampling-filtering method yields goodresults. The method includes the following successive steps:

Step 1: Generation of an oversampled frequency representation, ofdimensions N_(c)×N_(c) with for example N_(c)=4096. The array Fs is setto 0 and then the program goes through all the indices i,j, testing thecondition Ra[0,i,j]≠0. When the condition is true, it performs:${{Fs}\left\lbrack {{E\left( {{{Ia}\left\lbrack {0,0,i,j} \right\rbrack}\frac{N_{c}}{N_{pix}}} \right)},{E\left( {{{Ja}\left\lbrack {0,0,i,j} \right\rbrack}\frac{N_{c}}{N_{pix}}} \right)}} \right\rbrack} = {{Rb}\left\lbrack {0,i,j} \right\rbrack}$

where E(x) designates the integer number closest to x.

Step 2: An inverse Fourier transform is applied to array Fs

Step 3: The middle part of the array, of dimensions N_(pix)×N_(pix), isextracted and the presence of the diaphragm is simulated.${F_{rec}\left\lbrack {i,j} \right\rbrack} = {{Fs}\left\lbrack {{i - \frac{N_{pix}}{2} + \frac{N_{c}}{2}},{j - \frac{N_{pix}}{2} + \frac{N_{c}}{2}}} \right\rbrack}$

for all the pairs (i,j) such that${\left( {i - \frac{N_{pix}}{2}} \right)^{2} + \left( {j - \frac{N_{pix}}{2}} \right)^{2}} \leq \left( \frac{N_{pix}}{2} \right)^{2}$

where ouv is the aperture of the objectives.

Step 4: The Fourier transform of the array F_(rec) obtained isperformed.

7.11.3. Calculation of parameters

Array F_(rec) obtained after Step 4 constitutes the frequency image,equivalent to that whose acquisition is described in 7.8.1., butacquired in a manner less sensitive to local variations of the index. Itis possible to simulate this frequency image from a quintuplet (x,y,z,L,n₀) and to calculate a max characteristic value. This is what is done bythe part of the program described in FIG. 50 and including the steps(3509) to (3514), which finally calculates a max characteristic valuethat will be denoted max(x,y,z,L, n₀). The determination of the value ofthe quintuplet (x,y,z,L n₀) consists in using a maximization programwhich varies (x,y,z,L, n₀) so as to determine the point corresponding tothe highest value of max(x,y,z,L, n₀). In principle, any maximizationalgorithm is suitable. However, the number of variables and thecomplexity of the calculations are such that a specific and optimizedalgorithm must be used. The program described in 7.8. is thus used tocalculate the parameters from the array F_(rec) which constitutes thefrequency image.

7.12. Two-dimensional imagine

7.12.1. Principle

We saw in the first embodiment how it is possible to obtain atwo-dimensional frequency representation from several elementary imagesdiffering from each other in the phase difference between theilluminating beam and the reference beam, as well as in the attenuationlevel of the illuminating beam. In this embodiment, phase rotators(2210) (2238) (2241) (2226) are added which make it possible to vary thepolarization direction of the wave and the analysis direction. In fact,the diffraction of the illuminating wave is not homogeneous in alldirections. For a given direction, the diffracted wave depends to agreat extent on the polarization of the incident beam.

It will be recalled that if a and b are two vectors, then:

a{circumflex over ( )}b is the vector product of a and b

a.b is the scalar product of a and b

∥a∥ is the norm of a.

We shall call:

f_(c): characteristic frequency of a point

f_(e): illuminating frequency

f_(o): frequency associated with the optical center of the sensor

FIG. 51 shows the arrangement of these frequencies.

β_(c) is the angle between the vector f_(c){circumflex over ( )}f_(e)and the vector f_(c){circumflex over ( )}f_(o)

β_(e) is the angle between the vector f_(e){circumflex over ( )}f_(c)and the vector f_(e){circumflex over ( )}f_(o)

α_(c) is the angle between the vector {right arrow over (i)} and thevector f_(c){circumflex over ( )}f_(o)(the index p is omitted for thevectors {right arrow over (i)},{right arrow over (j)},{right arrow over(k)})

α_(e) is the angle between the vector {right arrow over (i)} and thevector f_(e){circumflex over ( )}f_(o)

θ is the angle between the vector f_(e) and the vector f_(c)

If the electric field vector of the illuminating beam is parallel, inthe object, to the vector f_(c){circumflex over ( )}f_(e), then the beamdiffracted by the object in the direction f_(c) is not attenuated. If itis orthogonal to this vector, the diffracted beam is attenuated by afactor cos θ.

The electric field vector of a beam, when it reaches the sensor, is inthe plane of the sensor, which is a plane orthogonal to f_(o). The modefor the passage from the electric field vector in the object to theelectric field vector on the sensor is a rotation around the vectorf_(c){circumflex over ( )}f_(o) for the frequency wave f_(c). Thisrotation preserves the angle β_(c). We denote (f_(c){circumflex over ()}f_(e))_(rc) the vector thus obtained from the vector f_(c){circumflexover ( )}f_(e). In the case of the frequency wave f_(e) polarizedparallel to f_(e){circumflex over ( )}f_(c), the rotation is aroundf_(e){circumflex over ( )}f_(o) and the resultant vector is denoted(f_(e){circumflex over ( )}f_(c))_(re)

We denote:${\overset{\rightarrow}{x}}_{c} = \frac{\left( {f_{c}\bigwedge f_{e}} \right)_{rc}}{{f_{c}\bigwedge f_{e}}}$${\overset{\rightarrow}{x}}_{e} = \frac{\left( {f_{e}\bigwedge f_{c}} \right)_{re}}{{f_{c}\bigwedge f_{e}}}$

 φ_(c)=α_(c)+β_(c)

φ_(e)=α_(e)+β_(e)

In the plane of the sensor, we thus have the configuration indicated inFIG. 52, in which the points O,E,C are respectively the optical center(frequency f_(o)), the point of direct impact of the illuminating wave(frequency f_(e)), and the point at which is measured the diffractedwave (frequency f_(c)).

When the electric field vector of the illuminating beam (at point E) isA₀{right arrow over (i)}+A₁{right arrow over (j)}, the resultingelectric field vector at point C is (C₀₀A₀+C₀₁A₁){right arrow over(i)}+(C₁₀A₀+C₁₁A₁){right arrow over (j)}.

During the measurements will be used:

an illuminating wave directed along {right arrow over (i)}. For thisilluminating wave will be measured the diffracted components polarizedalong the axes oriented by {right arrow over (i)} and {right arrow over(j)}, obtaining the factors C₀₀ and C₁₀

an illuminating wave directed along {right arrow over (j)}. For thisilluminating wave will be measured the diffracted components polarizedalong the axes oriented by {right arrow over (i)} and {right arrow over(j)}, obtaining the factors C₀₁ and C₁₁

The factors C_(kl) are thus the measured values.

We neglect here the fact that the polarization and analysis directionsobtained are not rigorously orthogonal.

If the electric field vector of the illuminating beam at point E isparallel to the vector {right arrow over (x)}_(e) and the electric fieldvector of the beam received at point C is parallel to the vector {rightarrow over (x)}_(c), there is no attenuation. This case corresponds toan electric field vector at point E of value A{right arrow over(x)}_(e)=A({right arrow over (i)} cos φ_(e)+{right arrow over (j)} sinφ_(e)) and hence a resulting electric field vector at point C: A[(C₀₀cos φ_(e)+C₀₁ sin φ_(e)){right arrow over (i)}+(C₁₀ cos φ_(e)+C₁₁ sinφ_(e)){right arrow over (j)}]. The projection of this vector on to theaxis oriented by −{right arrow over (x)}_(c) has the value: −A[(C₀₀ cosφ_(e)+C₀₁ sin φ_(e))cos φ_(c)+(C₁₀ cos φ_(e)+C₁₁ sin φ_(e))sin φ_(c)].The value to be integrated in the calculations is the ratio of thealgebraic value of the diffracted beam to that of the incident beam inthe absence of attenuation due to the angle θ, which thus corresponds tothe case above and is equal to: M=−cos φ_(e) cos φ_(c)C₀₀−sin φ_(e) cosφ_(c)C₀₁−cos φ_(e) sin φ_(c)C₁₀−sin φ_(e) sin φ_(c)C₁₁.

In order to calculate this value it is necessary to previously calculatethe functions cos φ_(e), cos φ_(c), sin φ_(e), sin φ_(c). This is doneby using, on the one hand, the trigonometric relations:

cos φ_(c)=cos α_(c) cos β_(c)−sin α_(c) sin β_(c)

sin φ_(c)=cos α_(c) sin β_(c)+sin α_(c) cos β_(c)

cos φ_(e)=cos α_(e) cos β_(e)−sin α_(e) sin β_(e)

sin φ_(e)=cos α_(e) sin β_(e)+sin α_(e) cos β_(e)

and, on the other hand, the following relations:${\cos \quad \beta_{c}} = {\frac{f_{c}\bigwedge f_{e}}{{f_{c}\bigwedge f_{e}}} \cdot \frac{f_{c}\bigwedge f_{o}}{{f_{c}\bigwedge f_{o}}}}$${\sin \quad \beta_{c}} = {\left( {\frac{f_{c}\bigwedge f_{o}}{{f_{c}\bigwedge f_{o}}}\bigwedge\frac{f_{c}\bigwedge f_{e}}{{f_{c}\bigwedge f_{e}}}} \right) \cdot \frac{f_{c}}{f_{c}}}$${\cos \quad \beta_{e}} = {\frac{f_{e}\bigwedge f_{c}}{{f_{e}\bigwedge f_{c}}} \cdot \frac{f_{e}\bigwedge f_{o}}{{f_{e}\bigwedge f_{o}}}}$${\sin \quad \beta_{e}} = {\left( {\frac{f_{e}\bigwedge f_{o}}{{f_{e}\bigwedge f_{o}}}\bigwedge\frac{f_{e}\bigwedge f_{c}}{{f_{e}\bigwedge f_{c}}}} \right) \cdot \frac{f_{e}}{f_{e}}}$${\cos \quad \alpha_{c}} = {\overset{\rightarrow}{i} \cdot \frac{f_{c}\bigwedge f_{o}}{{f_{c}\bigwedge f_{o}}}}$${\sin \quad \alpha_{c}} = {\overset{\rightarrow}{j} \cdot \frac{f_{c}\bigwedge f_{o}}{{f_{c}\bigwedge f_{o}}}}$${\cos \quad \alpha_{e}} = {\overset{\rightarrow}{i} \cdot \frac{f_{e}\bigwedge f_{o}}{{f_{e}\bigwedge f_{o}}}}$${\sin \quad \alpha_{e}} = {\overset{\rightarrow}{j} \cdot \frac{f_{e}\bigwedge f_{o}}{{f_{e}\bigwedge f_{o}}}}$

If we use normalized values of the frequency vectors:

f _(o) ={right arrow over (k)}

f _(e) =x _(e) {right arrow over (i)}+y _(e) {right arrow over (j)}+z_(e) {right arrow over (k)}with x _(e) ² +y _(e) ² +z _(e) ²=1

f _(c) =x _(c) {right arrow over (i)}+y _(c) {right arrow over (j)}+z_(c) {right arrow over (k)}with x _(c) ² +y _(c) ² +z _(c) ²=1

we obtain:${\cos \quad \phi_{c}} = {\frac{1}{M_{c}^{2}M_{ce}}\left( {{y_{c}^{2}V_{yz}} - {x_{c}y_{c}V_{xz}} + {x_{c}V_{xy}}} \right)}$${\sin \quad \phi_{c}} = {\frac{1}{M_{c}^{2}M_{ce}}\left( {{{- x_{c}}y_{c}V_{yz}} + {x_{c}^{2}V_{xz}} + {y_{c}V_{xy}}} \right)}$${\cos \quad \phi_{e}} = {{- \frac{1}{M_{e}^{2}M_{ce}}}\left( {{y_{e}^{2}V_{yz}} - {x_{e}y_{e}V_{xz}} + {x_{e}V_{xy}}} \right)}$${\sin \quad \phi_{e}} = {{- \frac{1}{M_{e}^{2}M_{ce}}}\left( {{{- x_{e}}y_{e}V_{yz}} + {x_{e}^{2}V_{xz}} + {y_{e}V_{xy}}} \right)}$

with:

V _(yz) =y _(c) z _(e) −z _(c) y _(e)

V _(xz) =−x _(c) z _(e) +z _(c) x _(e)

V _(xy) =−x _(c) y _(e) +y _(c) x _(e)

M _(c) ² =x _(c) ² +y _(c) ²

 M _(e) ² =x _(e) ² +y _(e) ²

M _(ce) ={square root over (V_(yz) ²+V_(xz) ²V_(xy) ²)}

However, when the denominators are zero, the above expressions must bereplaced by limit values.

7.12.2. Alaorithm

A two-dimensional frequency representation is an array M_(k,p,q)[i,j] ofcomplex numbers, with dimensions of N_(pix)×N_(pix) where k is therefractive index of the image in its series, p is the index of thesensor on which the direct illumination beam arrives (0 or 1) and q hasthe following values:

q=0: image received on the sensor on which the direct illuminating beamarrives

q=1: image received on the other sensor.

In addition to the image itself, this program also generates the arrays:

B_(k,p,q)[i,j]: noise indicator

H_(k,p,q)[i,j]: reference image, corresponding to a two-dimensionalfrequency representation obtained for a fixed direction of the beam. Thepurpose of this reference image is to subsequently allow thecompensation of optical path modifications due to vibrations in themirrors included in the system. If these vibrations are small, thereference image can be acquired only periodically. The reference imagemay also be a simplified image, the precision criteria being lower thanthose of the “useful” image M_(k,p,q)[i,j]. To simplify thepresentation, it will be supposed that a reference image H_(k,p,q)[i,j]having the same characteristics as the useful image (except for theilluminating beam direction) is acquired for each useful imageM_(k,p,q)[i,j].

BH_(k,p,q)[i,j]: reference image noise indicator.

A series of two-dimensional frequency representations is obtained bycalculation from a series of elementary images corresponding tointerference patterns formed on the CCD sensors. The two-dimensionalfrequency representations acquisition program is thus broken down intoan elementary image acquisition phase and a calculation phase. These twophases can be separated, or each image can form the subject of acalculation as the acquisition progresses. We are considering here thecase in which the two phases are separated.

7.12.2.1. Acquisition of elementary images

The series of elementary images can be acquired in a single operation bythe fast camera, without any calculations being carried out, in whichcase the control of the beam modification elements must be synchronizedwith image acquisition. This involves an iteration on the integer k andthe integer p. The succession of parallel illuminations to be used mustbe defined by arrays Ic[k,p], Jc[k,p] determining the “symbolic” indicesmaking it possible to calculate for each pair (k,p) the control word ofthe beam deflector COM[p,Ic[k,p],Jc[k,p]]. During this procedure, thebeams FRG and FRD are present at all times, each elementary image beingformed on a sensor by the interference of the reference wave and thewave diffracted by the sample, itself illuminated by illuminating wavesof variable characteristics. The mirrors (2282) (2243) (2232) (2247) areblocked.

For each pair (k,p), imaging is broken down into two phases:

Phase 1: Acquisition of 36 pairs of elementary images, a pair of imagescomprising an image coming from each sensor, and the 36 pairs differingfrom each other in the condition of all the beam control systems exceptfor the beam deflector which maintains a constant condition for a givenpair (k,p). These 36 pairs of elementary images will be used later on togenerate a useful two-dimensional frequency representationM_(k,p,q)[i,j]. One of these elementary images is denotedM0[k,p][c,d,r₁,r₂][q,i,j]. Before each elementary imaging operation, thephase rotators allowing the control of the illuminating beam(attenuation, phase shift, deflection and polarization) must becontrolled appropriately.

The index c is determined by the following table, in which att[c]constitutes a table containing the attenuation corresponding to theindex, and in which the values a₁ and a₂ are those explained in 7.2.2.,measured in 7.3.2.2., and in which the beam attenuator is controlled asexplained in 7.2.2.

Index c Attenuation att[c] 0 1 1 α₁ 2 α₁ α₂

The indices d,r₁,r₂ are determined by the following tables:

Index d Phase shift (degrees) 0 +120 1 0 2 −120 Voltage applied to therotators (2210) and Index r₂ (2241) 0   5 V 1 −5 V Voltage applied tothe rotators (2238) and Index r₁ (2226) 0   5 V 1 −5 V

A pair of elementary images (corresponding to the two values of theindex q) is obtained for each combination of indices c,d,r₁,r₂ and foreach illuminating direction defined by the indices (k,p). For eachilluminating direction (k,p), one thus obtained 36 pairs of elementaryimages denoted M0[k,p][c,d,r₁,r₂][q,i,j].

The filter (2203) must be adjusted so that the sensor is neversaturated, but comes as close as possible to the saturation limit.Equivalently, it may be adjusted in the absence of a reference wave sothat the maximum intensity of the wave reaching the sensor in theabsence of beam attenuation is one-fourth the maximum level authorizedby the digitization of the video signal, i.e. 64 for an 6-bit digitizer.

Phase 2: Acquisition of 36 additional elementary images which will beused to generate the reference image. This phase is identical to Phase 1but:

we shall denote MR0[k,p][c,d,r₁,r₂][q,i,j] an elementary image obtained.

the control word used to obtain MR0[k,p][c,d,r₁,r₂][q,i,j] isCOM[p,Id[p,i_(r),j_(r)], Jd[p,i_(r),j_(r)]] in which i_(r),j_(r) are thecoordinates of a constant point located on the side of the sensor, forexample the point (3905) in FIG. 56, representing a sensor and on whichthe contour (3902) represents the limit corresponding to the aperture ofthe objective, which cannot be exceeded by the illuminating beams. Thiscontrol word consequently does not depend on k. The coordinatesi_(r),j_(r) can be chosen in a relatively arbitrary manner. However, thechoice of a highly eccentric point (3905) makes it possible for thepoints corresponding to total frequency vectors (f_(t) according to thenotations used in 5.3.) of comparable standard to be obtained from eachsensor. For samples not having specific regular structures, such pointscorrespond to comparable complex values on the two sensors, thusimproving the reliability of the results.

7.12.2.2. Calculation of two-dimensional frequency representations

After the acquisition phase, a specific program must be used togenerate, from these elementary images, images in complex numbers andthe associated noise indicators. To generate M_(k,p,q)[i,j] andB_(k,p,q)[i,j], the program goes through the indices (k,p), carrying outfor each pair (k,p) the following three steps:

Step 1: Generation of the attenuation indicator array.

This array is an array of integers M1[q,i,j] generated as follows: it isinitialized to 0, then the program goes through all the indices q,i,jtwice.

First passage: the program calculates${\max_{—}{pix}} = {\max\limits_{d,r_{1},r_{2}}{\left( {{{{M0}\left\lbrack {k,p} \right\rbrack}\left\lbrack {0,{dr}_{1},r_{2}} \right\rbrack}\left\lbrack {q,i,j} \right\rbrack} \right).}}$

If, for a given value of q,i,j, max_pix is equal to the maximum value ofthe digitizer, the program sets this pixel and its immediate neighborsto 1:

M1[q,i+iadd,j+jadd]=1 for iadd and jadd varying from −1 to 1, andprovided that the limits of the array are not exceeded.

Second passage: the program calculates${\max_{—}{pix}} = {\max\limits_{d,r_{1},r_{2}}{\left( {{{{M0}\left\lbrack {k,p} \right\rbrack}\left\lbrack {0,{dr}_{1},r_{2}} \right\rbrack}\left\lbrack {q,i,j} \right\rbrack} \right).}}$

If, for a given value of q,i,j, max_pix is equal to the maximum value ofthe digitizer, the program sets this pixel and its immediate neighborsto 2:

M1[q,i+iadd,j+jadd]=2 for iadd andjadd varying from −1 to 1, andprovided the limits of the array are not exceeded.

After this step, the array M1[q,i,j] contains the index corresponding tothe attenuation to be used to attenuate the image.

Step 2: Generation of complex images corresponding to each position ofthe rotators.${{{M2}\left\lbrack {r_{1},r_{2}} \right\rbrack}\left\lbrack {q,i,j} \right\rbrack} = {\frac{1}{{att}\left\lbrack {{M1}\left\lbrack {q,i,j} \right\rbrack} \right.}\left\lbrack {{\frac{1}{6\sqrt{{Iref}\left\lbrack {{{\overset{\_}{p}q} + {p\overset{\_}{q}}},i,j} \right\rbrack}}\left( {{2{{{{M0}\left\lbrack {k,p} \right\rbrack}\left\lbrack {{{M1}\left\lbrack {q,i,j} \right\rbrack},0,r_{1},r_{2}} \right\rbrack}\left\lbrack {q,i,j} \right\rbrack}} - {{{{M0}\left\lbrack {k,p} \right\rbrack}\left\lbrack {{{M1}\left\lbrack {q,i,j} \right\rbrack},1,r_{1},r_{2}} \right\rbrack}\left\lbrack {q,i,j} \right\rbrack} - {{{{M0}\left\lbrack {k,p} \right\rbrack}\left\lbrack {{{M1}\left\lbrack {q,i,j} \right\rbrack},2,r_{1},r_{2}} \right\rbrack}\left\lbrack {q,i,j} \right\rbrack}} \right)} + {\overset{\sim}{j}\frac{1}{2\sqrt{3\quad {{Iref}\left\lbrack {{{\overset{\_}{p}q} + {p\overset{\_}{q}}},i,j} \right\rbrack}}}\left( {{{{{M0}\left\lbrack {k,p} \right\rbrack}\left\lbrack {{{M1}\left\lbrack {q,i,j} \right\rbrack},1,r_{1},r_{2}} \right\rbrack}\left\lbrack {q,i,j} \right\rbrack} - {{{{M0}\left\lbrack {k,p} \right\rbrack}\left\lbrack {{{M1}\left\lbrack {q,i,j} \right\rbrack},2,r_{1},r_{2}} \right\rbrack}\left\lbrack {q,i,j} \right\rbrack}} \right)}} \right\rbrack}$

Step 3: Combination of images obtained for the various positions of therotators. The purpose of Step 3 is to calculate M_(k,p,q)[i,j] as afunction of M2[r₁,r₂][q,i,j]. This can be achieved simply without usingthe polarization variations by performing: M_(k,p,q)[i,j]=M2[1,1][q,i,j]the noise then being given by:${B_{k,p,q}\left\lbrack {i,j} \right\rbrack} = \frac{1}{{att}\left\lbrack {{M1}\left\lbrack {q,i,j} \right\rbrack} \right\rbrack}$

The value M_(k,p,q)[i,j] thus generated corresponds to the one used inthe first embodiment. Nevertheless, this method induces inaccuracies onthe high frequencies and it is preferable to use the principle describedin 7.12.1., in which M2[r₁,r₂][q,i,j] corresponds to the measured value,which was denoted C_(r) ₁ _(,r) ₂ , in 7.12.1. Other variants of thismethod will be described in 7.18.

Step 3 is thus carried out as follows:

For each value of the indices q,i,j, the program calculates:

po=p{overscore (q)}+{overscore (p)}q

${x_{c} = \frac{i - \frac{N_{pix}}{2}}{\frac{n_{o}}{n_{v}}K_{po}}},{y_{c} = \frac{j - \frac{N_{pix}}{2}}{\frac{n_{o}}{n_{v}}K_{po}}},{z_{c} = \sqrt{1 - x_{c}^{2} - y_{c}^{2}}}$${x_{e} = \frac{{{la}\left\lbrack {q,p,{{Ic}\left\lbrack {k,p} \right\rbrack},{{Jc}\left\lbrack {k,p} \right\rbrack}} \right\rbrack} - \frac{N_{pix}}{2}}{\frac{n_{o}}{n_{v}}K_{po}}},{y_{e} = \frac{{{Ja}\left\lbrack {q,p,{{Ic}\left\lbrack {k,p} \right\rbrack},{{Jc}\left\lbrack {k,p} \right\rbrack}} \right\rbrack} - \frac{N_{pix}}{2}}{\frac{n_{o}}{n_{v}}K_{po}}},{z_{e} = \sqrt{1 - x_{e}^{2} - y_{e}^{2}}}$

 V _(yz) =y _(c) z _(e) −z _(c) y _(e)

V _(xz) =−x _(c) z _(e) +z _(c) x _(e)

V _(xy) =−x _(c) y _(e) +y _(c) x _(e)

M _(c) ² =x _(c) ² +y _(c) ²

M _(e) ² =x _(e) ² +y _(e) ²

M _(ce) ={square root over (V_(yz) ²+V_(xz) ²+V_(xy) ²)}

The values of sin φ_(c) cos φ_(c) sin φ_(e) cos φ_(e) are determinedaccording to the following tables:

M_(ce) 0 other M_(c) 0 other cosφ_(c) 1 $- \frac{y_{e}}{M_{e}}$

$\frac{1}{M_{c}^{2}M_{ce}}\left( {{y_{c}^{2}V_{yz}} - {x_{c}y_{c}V_{xz}} + {x_{c}V_{xy}}} \right)$

sinφ_(c) 0 $\frac{x_{e}}{M_{e}}$

$\frac{1}{M_{c}^{2}M_{ce}}\left( {{{- x_{c}}y_{c}V_{yz}} + {x_{c}^{2}V_{xz}} + {y_{c}V_{xy}}} \right)$

M_(ce) 0 other M_(e) 0 other cosφ_(e) −1  $- \frac{y_{c}}{M_{c}}$

${- \frac{1}{M_{e}^{2}M_{ce}}}\left( {{y_{e}^{2}V_{yz}} - {x_{e}y_{e}V_{xz}} + {x_{e}V_{xy}}} \right)$

sinφ_(e) 0 $\frac{x_{c}}{M_{c}}$

${- \frac{1}{M_{e}^{2}M_{ce}}}\left( {{{- x_{e}}y_{e}V_{yz}} + {x_{e}^{2}V_{xz}} + {y_{e}V_{xy}}} \right)$

Then the coefficients are calculated:

coef[k,p,q,i,j][0,0]=−cos φ_(e) cos φ_(c)

coef[k,p,q,i,j][0,1]=−sin φ_(e) cos φ_(c)

coef[k,p,q,i,j][1,0]=−cos φ_(e) sin φ_(c)

coef[k,p,q,i,j][1,1]=−sin φ_(e) sin φ_(c)

These coefficients do not depend on the result of the imaging and, ifthe same series is always repeated, they may be stored in an arrayrather than be calculated again each time. That is why they have beenexpressed here in this form. The program then uses these values tocombine the images obtained with the different positions of the rotatorsas follows:${M_{k,p,q}\left\lbrack {i,j} \right\rbrack} = {\sum\limits_{r_{1},r_{2}}{{{{M2}\left\lbrack {r_{1},r_{2}} \right\rbrack}\left\lbrack {q,i,j} \right\rbrack}{{{coef}\left\lbrack {k,p,q,i,j} \right\rbrack}\left\lbrack {r_{1},r_{2}} \right\rbrack}}}$

M2[r₁,r₂][q,i,j] corresponds to the measured value, which was denotedC_(r) ₁ _(,r) ₂ in 7.12.1

M_(k,p,q)[i,j] corresponds to the value which was denoted M in 7.12.1

Moreover, the program calculates a noise amplitude:${B_{k,p,q}\left\lbrack {i,j} \right\rbrack} = \frac{1}{{att}\left\lbrack {{M1}\left\lbrack {q,i,j} \right\rbrack} \right\rbrack}$

This ends Step 3.

When the program has calculated M_(k,p,q)[i,j] and B_(k,p,q)[i,j], italso calculates H_(k,p,q)[i,j] and BH_(k,p,q)[i,j]. This secondcalculation is carried out in the same manner as the preceding one, inthree steps, but M0[k,p][c,d,r₁,r₂][q,i,j], M_(k,p,q)[i,j] andB_(k,p,q)[i,j] are replaced respectively by MR0[k,p][c,d,r₁,r₂][q,i,j],H_(k,p,q)[i,j] and BH_(k,p,q)[i,j] and, in Step 3, the values of x_(e)and y_(e) are replaced by${x_{e} = \frac{i_{r} - \frac{N_{pix}}{2}}{\frac{n_{o}}{n_{v}}K_{po}}},{y_{e} = \frac{j_{r} - \frac{N_{pix}}{2}}{\frac{n_{o}}{n_{v}}K_{po}}}$

The above procedure is the one that offers the maximum precision.However, owing to the large number of elementary images required, it maybe necessary to use a faster procedure. It is possible not to use thefour images generated by the combination of the indices r₁, r₂, as inthe first embodiment in which a single image is generated. It is alsopossible not to use beam attenuation. The fastest method consists inusing only three elementary images differing from each other in theirphase. The reference image can also be acquired only one out of everyten times (for example) in order to limit the waste of time related toits acquisition, and provided that the vibrations are not too large. Thereference image can be simplified just like the useful image, but thissimplification will generally have a smaller impact on the quality ofthe results. It is thus not useful to simplify further the useful image.

7.13. Calculation of control indices

The array of control indices is the array Id which makes it possible toassociate, with coordinates in pixels (i,j) and with a sensor p, virtualcontrol indices (Id[p,i,j], Jd[p,i,j]) such that the control wordCOM(p,Id[p,i,j],Jd[p,i,j]) generates an illumination illuminating apoint as close as possible to the point having coordinates (i,j) on thesensor p. This array is generated by the algorithm in FOG. 53. In thisalgorithm, E(x) designates the integer as close as possible to x. Beforestarting this algorithm, the array D must be initialized at a highvalue, for example 100000. After this program, the array D contains, foreach point, the distance between this point and the nearest pointobtained for the point of direct impact of an illuminating wave. Thedouble loop of the algorithm, on (i₀,j₀) and on (i₁,j₁), allows thedefinition of values Id[p,i,j],Jd[p,i,j], also for points which do notcorrespond exactly to the point of direct impact of the beam.

A trajectory for the point of direct impact can be defined by the arrays(Io[k],Jo[k]) defining, as a function of the index k, the coordinates ofthe desired point of direct impact. For example, if the trajectory is acircle of radius R pixels, it is possible to have: Io[k]=R cos k/R ,Jo[k]=R sin k/R for k ranging from 0 to 2πR.

The arrays defining the control indices as a function of k and p areobtained from Io[k] and Jo[k] by:

Ic[k,p]=Id[p,Io[k],Jo[k]] and Jc[k,p]=Jd[p,Io[k],Jo[k]]

7.14. Path difference induced on waves coming from the object

FIG. 54 illustrates the calculation of the path difference of a wavecoming from a point O of the object in relation to the reference wavecoming virtually from a point A of a medium of index n_(v) (nominalrefractive index of the objectives). We have:

Δ=n ₀ l ₀ −n _(v) l _(v)

Δ=n ₀ d ₀ cos β−n _(v) d _(v) cos α

where n₀ sin β=n_(v) sin α

giving, finally:$\Delta = {{n_{0}\quad d_{0}\sqrt{1 - \left( {\frac{n_{v}}{n_{0}}\quad \sin \quad \alpha} \right)^{2}}} - {n_{v}\quad d_{v}\sqrt{1 - {\sin^{2}\quad \alpha}}}}$

FIG. 55 illustrates the calculation of the path difference between awave coming from a point O of the object and the reference wave comingvirtually from the point A_(p) where p is the index of the consideredsensor. The coordinates of A_(p) in relation to a coordinate systemcentered on O and direction vectors {right arrow over (i)}_(p),{rightarrow over (j)}p,{right arrow over (k)}_(p) are x_(p),y_(p),z_(p), andthe distance between O and the edge of the object on the side of thesensor p is w_(p). The vectors {right arrow over (i)}_(p),{right arrowover (j)}_(p),{right arrow over (k)}_(p) are defined as indicated in7.7. We verify that, in accordance with the orientation of the axes inFIG. 55, the base vectors of the coordinate systems used in eachhalf-space referenced by the index of sensor p comply with: {right arrowover (i)}₀=−{right arrow over (i)}₁,{right arrow over (j)}₀=−{rightarrow over (j)}₁,{right arrow over (k)}₀=−{right arrow over (k)}₁.

When x_(p)=y_(p)=0, it is possible to apply the preceding formula:$\Delta = {{n_{0}\quad d_{0}\sqrt{1 - \left( {\frac{n_{v}}{n_{0}}\quad \sin \quad \alpha} \right)^{2}}} - {n_{v}\quad d_{v}\sqrt{1 - {\sin^{2}\quad \alpha}}}}$

where

 d ₀ =w _(p)

d _(v) =w _(p) −z _(p)${\sin^{2}\quad \alpha} = \frac{i^{2} + j^{2}}{K_{p}^{2}}$

where i,j are the coordinates in pixels taken from the optical center ofthe sensor.

If x_(p), y_(p) are also taken into account, to this path differencemust be added the quantity:$n_{v}\quad \left( {{x_{p}\quad \frac{i}{K_{p}}} + {y_{p}\quad \frac{j}{K_{p}}}} \right)$

and one finally obtains, for the total path difference:$\left( {x_{p},y_{p},z_{p}} \right) = \left( {\frac{x}{2},\frac{y}{2},\frac{z}{2}} \right)$

In particular, it is possible to position the point O so as to have

(x _(p) ,y _(p) ,z _(p))=(x/2,y/2,z/2)

where x,y,z are the coordinates in 7.11. We then have:$\Delta_{p} = {{n_{0}\quad w_{p}\sqrt{1 - {\left( \frac{n_{v}}{n_{0}} \right)^{2}\quad \frac{i^{2} + j^{2}}{K_{p}^{2}}}}} - {{n_{v}\left( {w_{p} - \frac{z}{2}} \right)}\sqrt{1 - \frac{i^{2} + j^{2}}{K_{p}^{2}}}} + {n_{v}\left( {{\frac{x}{2}\quad \frac{i}{K_{p}}} + {\frac{y}{2}\quad \frac{j}{K_{p}}}} \right)}}$

To obtain a frequency representation of the object, the two-dimensionalfrequency representations must be corrected to compensate for this pathdifference. In this expression, only the values of w_(p) have not yetbeen determined.

7.15. Calculation of w_(p).

7.15.1. Principle

To correct the two-dimensional frequency representations of the phasefactor determined in 7.14., it is necessary to previously determine thevalues w_(p), i.e. in fact only the value w₀ since w₁ is deducedtherefrom through w₁=L−w₀.

If the average index of the object is close to the nominal index of theobjectives, the effect of w_(p) on the value Δ_(p) is negligible and itis possible, for example, to adopt the value w_(p)=L/2 and position thesample between the two objectives visually, by adjusting this positionsubsequently to obtain an image of the region of interest of thissample.

It is also possible to add a reflecting layer on the side of one of thecover glasses which is in contact with the object, for example the onelocated on the side of the objective (2217), in a region of reduceddimensions. When the beam FRD and its opposite indicator are used aloneand when the reflecting part is positioned so as to reflect the oppositeindicator beam of FRD, then the interference pattern formed on thesensor (2239) must be a constant. The position of the sample must thenbe adjusted to actually obtain such a constant. When this adjustment hasbeen performed, we have w₀=z/2. If a sufficiently precise positioner isused, the position of the sample can then be modified in the directionof the optical axis to obtain w₀=L/2. The position of the sample mustfinally be modified in the direction orthogonal to the optical axis sothat the reflecting zone of the cover glass is located outside the fieldof observation.

However, the two preceding methods impose practical constraints whichmay be troublesome. A solution making it possible to avoid thisdifficulty is the determination of w_(p) from measurements carried outon the sample in its final position.

A frequency representation Fa can be obtained from all thetwo-dimensional frequency representations coming from the sensor 0(2239) when the point of direct impact of the illuminating wave is onthis same sensor and takes, on this sensor, for example, the pathrepresented in broken lines in FIG. 56. This representation is obtainedin a manner very similar to the method used in the first embodiment,however with the following differences:

The two-dimensional frequency representation must be multiplied by thecorrection factor$^{{- j}\quad 2\quad \pi \quad \frac{\Delta_{p}}{\lambda}}$

 to cancel the phase difference produced by spherical aberration.

The value of the coefficient K taken into account must be multiplied bya factor n_(o)/n_(v) to take account of the average index in the sample.

To limit aliasing of the correction function, the frequencyrepresentation is oversampled.

When a point of the space of the frequencies is obtained several times,the adopted value is one of the values obtained and not the average ofthe values obtained.

In principle, the frequency representation thus obtained does not dependon the choice of the value adopted when a point of the frequency spaceis obtained several times. However, this is true only if the correctionfactor has a correct value.

Under these conditions, and taking into account the expression of thecorrection function$^{{- j}\quad 2\quad \pi \quad \frac{\Delta_{p}}{\lambda}},$

the value of the two-dimensional frequency representation Fa thusobtained at a point of the frequency space of coordinates ni,nj,nk, canbe written in the following form:

Fa[ni,nj,nk]=Fs[ni,nj,nk]exp(−j2πG[ni,nj,nk]w ₀)

When a point of the frequency space is obtained several times, thevalues Fs[ni,nj,nk] and G[ni,nj,nk] obtained are different each time.For each point, we define Gmin[ni,nj,nk] and Gmax[ni,nj,nk], minimum andmaximum values obtained for G at this point. Fsmin[ni,ni,nk] andFsmax[ni,nj,nk] are then the values obtained for Fs[ni,nj,nk] whenG[ni,nj,nk] is equal respectively to Gmin[ni,nj,nk] and Gmax[ni,nj,nk].

The following two frequency representations are then obtained:

Famin[ni,nj,nk]=Fsmin[ni,nj,nk]exp(−j2πGmin[ni,nj,nk]w ₀)

Famax[ni,nj,nk]=Fsmax[ni,nj,nk]exp(−j2πGmin[ni,nj,nk]w ₀)

When the value of w₀ is correct, these two representations are equal.The calculation of w₀ consists in minimizing the standard deviationbetween the two frequency representations Famin and Famax. The standarddeviation to be minimized is in principle:${ecart} = {\sum\limits_{{ni},{nj},{nk}}^{\quad}\quad {{{{Famax}\left\lbrack {{ni},{nj},{nk}} \right\rbrack} - {{Famin}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}}}^{2}}$

However, as the noise is not constant over the entire frequencyrepresentation, each element of this sum must be weighted by the inverseof the noise at the considered point and we obtain:${ecart} = {\sum\limits_{{ni},{nj},{nk}}^{\quad}\quad {\frac{{{Famax}\left\lbrack {{ni},{nj},{nk}} \right\rbrack} - {{Famin}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}}{{Btot}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}}^{2}}$

where Btot[ni,nj,nk] is a noise amplitude defined at each point.

By developing the expression of this standard deviation, a simplifiedexpression is obtained which facilitates the minimization calculation.

7.15. 2. Algorithm

The calculation of w_(p) is carried out with a program whose algorithmis described in FIG. 57. This program initially requires the followinginformation:

values established in 7.11.: x,y,z,L, n₀

operating parameter wpixels, for example wpixels=5.

The essential steps in this program are the following:

(4001): image acquisition. A series of images is obtained by having thepoint of direct impact of the illuminating beam take the pathrepresented in broken lines in FIG. 56, where (3901) represents thelimit of the useful region of the sensor, (3902) represents the limitcorresponding to the maximum aperture of the objectives, (3903)represents a circular part of the path and (3904) represents a straightpart of the path. As indicated in 7.13., the arrays Io[k] Jo[k] aregenerated along this path and the program calculates the correspondingcontrol indices Ic[k,p] Jc[k,p]. The program then acquires the series ofimages according to the procedure indicated in 7.12. The acquisitionprocedure generates the arrays M_(k,p,q)[i,j]. However, only the arrayscorresponding to zero indices q,p will be used here, i.e.M_(k,0,0)[i,j].

(4002): calculation of Fsmin, Gmin, Fsmax, Gmax, Btot

Step 1: Each array M_(k,0,0)[i,j] obtained during the acquisition phaseis first oversampled as follows in three steps:

Step 1.1.: the program takes the two-dimensional inverse Fouriertransform of the array M_(k,0,0)

Step 1.2.: the program completes this array by zeros to obtain an arrayMs_(k) of dimensions N_(d)×N_(d) (for example N_(d)=512). Ms_(k) isinitialized to 0 then the program goes through the indices i,j of thearray M_(k,0,0)performing:${{Ms}_{k}\left\lbrack {\frac{N_{d}}{2} + i - \frac{N_{pix}}{2}} \right\rbrack} = {M_{k,0,0}\left\lbrack {i,j} \right\rbrack}$

Step 1.3.: a direct Fourier transform of the array Ms_(k) is taken, thusterminating the oversampling phase.

Step 2: The elements of each array Ms_(k) are referred to the valueobtained at the point of direct impact of the illuminating wave:${{Ms}_{k}\left\lbrack {i,j} \right\rbrack} = \frac{{Ms}_{k}\left\lbrack {i,j} \right\rbrack}{{Ms}_{k}\left\lbrack {{imax}_{k},{jmax}_{k}} \right\rbrack}$

 where we denote:${imax}_{k} = {E\quad \left( {\frac{N_{d}}{N_{pix}}\quad {{Ia}\left\lbrack {0,0,{{Ic}\left\lbrack {k,0} \right\rbrack},{{Jc}\left\lbrack {k,0} \right\rbrack}} \right\rbrack}} \right)}$${jmax}_{k} = {E\quad \left( {\frac{N_{d}}{N_{pix}}\quad {{Ja}\left\lbrack {0,0,{{Ic}\left\lbrack {k,0} \right\rbrack},{{Jc}\left\lbrack {k,0} \right\rbrack}} \right\rbrack}} \right)}$

where E(x) designates the integer closest to x.

Step 3: When the program has thus generated all the oversampled arraysMs_(k), it calculates an oversampled noise amplitude in the form of aset of arrays of positive real values Bs_(k) with dimensionsN_(d)×N_(d). For this purpose, it goes through the indices i,j,k where iand j vary between 0 and N_(d)−1 by performing:${{Bs}_{k}\left\lbrack {i,j} \right\rbrack} = {{\max\limits_{\underset{{{j/r} - 2} \leq j_{1} \leq {{j/r} + 2}}{{{i/r} - 2} \leq i_{1} \leq {{i/r} + 2}}}\quad {\left( {B_{k,0,0}\left\lbrack {i_{1},j_{1}} \right\rbrack} \right)\quad {{where}:r}}} = \frac{N_{d}}{N_{pix}}}$

In this equation, when the pair i₁,j₁ is outside the limits of the arrayB_(k,0,0) the coefficient B_(k,0,0)[i₁,j₁] is assumed to be equal to 0.

Step 4: The program initializes to 0 the arrays Fsmin, Fsmax; itinitializes respectively to 10²⁰ and −10²⁰ the arrays Gmin and Gmax; andinitializes to 10²⁰ the arrays Bmin,Bmax. These arrays each have thedimensions 2 N_(d)×2 N_(d)×2 N_(d).

The program then goes through the indices i,j,k where i and j vary from0 to N_(d)−1, performing for each triplet (i,j,k) the following steps,numbered from 4.1 to 4.3:

Step 4.1.: calculation of indices of the three-dimensional frequencyrepresentation. The program performs the following operations:

 ni=i−imax_(k) +N _(d)

nj=j−jmax_(k) +N _(d)

${nk} = {\sqrt{\left( {r\frac{n_{o}}{n_{v}}K_{0}} \right)^{2} - \left( {i - \frac{N_{d}}{2}} \right)^{2} - \left( {j - \frac{N_{d}}{2}} \right)^{2}} - \sqrt{\left( {r\frac{n_{o}}{n_{v}}K_{0}} \right)^{2} - \left( {{imax}_{k} - \frac{N_{d}}{2}} \right)^{2} - \left( {{jmax}_{k} - \frac{N_{d}}{2}} \right)^{2}} + N_{d}}$

where: $r = \frac{N_{d}}{N_{pix}}$

The values ni,nj,nk correspond to coordinates in a three-dimensionalfrequency space as in the first embodiment. As their calculation leadsto non-integer values, the integer nearest the obtained value isassigned to them. The value of K must be corrected to take into accountthe index of the sample and we thus use n_(o)/n_(v)K₀. The coefficient rmakes it possible to take into account the oversampling.

Step 4.2.: calculations of values of G and Fs at the current point:${Gval} = {\frac{1}{\lambda}\left\{ {\left( {{n_{0}\sqrt{1 - {\left( \frac{n_{v}}{n_{0}} \right)^{2}\quad \frac{{ic}^{2} + {jc}^{2}}{r^{2}K_{0}^{2}}}}} - {n_{v}\sqrt{1 - \frac{{ic}^{2} + {jc}^{2}}{r^{2}K_{0}^{2}}}}} \right) - \left( {{n_{0}\sqrt{1 - {\left( \frac{n_{v}}{n_{0}} \right)^{2}\frac{{imc}^{2} + {jmc}^{2}}{r^{2}K_{0}^{2}}}}} - {n_{v}\sqrt{1 - \frac{{imc}^{2} + {jmc}^{2}}{r^{2}K_{0}^{2}}}}} \right)} \right\}}$${Fsval} = {{{Ms}_{k}\left\lbrack {i,j} \right\rbrack}^{{- \overset{\sim}{j}}\quad \frac{2\quad \pi}{\lambda}n_{v}{\{{{({{\frac{z}{2}\sqrt{1 - \frac{{ic}^{2} + {jc}^{2}}{r^{2}K_{0}^{2}}}} + {\frac{x}{2}\frac{ic}{{rK}_{0}}} + {\frac{y}{2}\quad \frac{jc}{{rK}_{0}}}})} - {({{\frac{z}{2}\sqrt{1 - \frac{{imc}^{2} + {jmc}^{2}}{r^{2}K_{0}^{2}}}} + {\frac{x}{2}\frac{imc}{{rK}_{0}}} + {\frac{y}{2}\quad \frac{jmc}{{rK}_{0}}}})}}\}}}}$

where:${{ic} = {i - \frac{N_{d}}{2}}},{{jc} = {j - \frac{N_{d}}{2}}},{{imc} = {{imax}_{k} - \frac{N_{d}}{2}}},{{jmc} = {{jmax}_{k} - \frac{N_{d}}{2}}}$

In the above equations, ic,jc,imc,jmc correspond to coordinates referredto the optical center and r has the same value as in the previous step.

Step 4.3.: possible modification of the values of Gmin,Fsmin,Gmax,Fsmax.

The program tests the value of Gval.

If Gval≦Gmin[ni,nj,nk] the program carries out:

Gmin[ni,nj,nk]=Gval

Fsmin[ni,nj,nk]=Fsval

Bmin[ni,nj,nk]=Bs _(k) [i,j]

If Gval≧Gmaxn[ni,nj,nk] the program carries out:

Gmax[ni,nj,nk]32 Gval

Fsmax[ni,nj,nk]=Fsval

Bmax[ni,nj,nk]=BS _(k) [i,j]

Step 5: The program generates an overall noise amplitude. For thispurpose, it goes through the indices ni,nj,nk, performing for each ofthese triplets, the operation:

Btot[ni,nj,nk]={square root over (|Bmax[ni,nj,nk]|²+|B min[ni,nj,nk]|²)}

(4003): The calculated function is in principle equal to:${escart} = {\sum\limits_{{({{ni},{nj},{nk}})} \in {Es}}\left| \frac{\begin{matrix}{{{{Fsmin}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}^{{- \overset{\sim}{j}}2\pi \quad {{Gmin}{\lbrack{{ni},{nj},{nk}}\rbrack}}w_{0}}} -} \\{{{Fsmax}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}^{{- \overset{\sim}{j}}2\pi \quad {{Gmax}{\lbrack{{ni},{nj},{nk}}\rbrack}}w_{0}}}\end{matrix}}{{Btot}\left\lbrack {{ni},{nj},{nk}} \right\rbrack} \right|^{2}}$

where Es is the set of points at which the values of Gmin and Gmaxdiffer, namely:

Es={ni,nj,nk|Gmax[ni,nj,nk]≠Gmin[ni,nj,nk]}

Hence, developing the expression:${ecart} = {\sum\limits_{{({{ni},{nj},{nk}})} \in {Es}}\left| \frac{{Fsmin}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}{{Btot}\left\lbrack {{ni},{nj},{nk}} \right\rbrack} \middle| {}_{2}{+ \left| \frac{{Fsmax}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}{{Btot}\left\lbrack {{ni},{nj},{nk}} \right\rbrack} \middle| {}_{2}\quad {{- \quad \quad 2}\quad {\sum\limits_{{({{ni},{nj},{nk}})} \in {Es}}{{Re}\left( {\frac{{{Fsmin}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}\overset{\_}{{Fsmax}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}}{\left| {{Btot}\left\lbrack {{ni},{nj},{nk}} \right\rbrack} \right|^{2}}\quad ^{{- \overset{\sim}{j}}2\quad {\pi {({{{Gmin}{\lbrack{{ni},{nj},{nk}}\rbrack}} - {{Gmax}{\lbrack{{ni},{nj},{nk}}\rbrack}}})}}w_{0}}} \right)}}} \right.} \right.}$

The first part of the expression does not depend on w₀. Minimizing thestandard deviation is thus equivalent to maximizing the followingfunction in which Re( ) designates the real part:$\sum\limits_{{({{ni},{nj},{nk}})} \in {Es}}{{Re}\left( \quad {\frac{{{Fsmin}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}\overset{\_}{{Fsmax}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}}{\left| {{Btot}\left\lbrack {{ni},{nj},{nk}} \right\rbrack} \right|^{2}}^{{- \overset{\sim}{j}}2\quad {\pi {({{{Gmin}{\lbrack{{ni},{nj},{nk}}\rbrack}} - {{Gmax}{\lbrack{{ni},{nj},{nk}}\rbrack}}})}}w_{0}}} \right)}$

However, this function exhibits high frequencies causing aliasing whichdisturb the convergence of the algorithm. They are eliminated by usingthe function:${f\left( {w,{\Delta \quad w}} \right)} = {\sum\limits_{{({{ni},{nj},{nk}})} \in {Es}}\left\{ {{{Re}\left( \quad {\frac{{{Fsmin}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}\overset{\_}{{Fsmax}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}}{\left| {{Btot}\left\lbrack {{ni},{nj},{nk}} \right\rbrack} \right|^{2}}{\exp \left( {{- \overset{\sim}{j}}2\quad {\pi \left( {{{Gmin}\left\lbrack {{ni},{nj},{nk}} \right\rbrack} - {{Gmax}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}} \right)}w_{0}} \right)}} \right)}{\gamma \left( {\left( {{{Gmin}\left\lbrack {{ni},{nj},{nk}} \right\rbrack} - {{Gmax}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}} \right)\Delta \quad w} \right)}} \right\}}$

${{where}\quad {\gamma (x)}} = {\left. {0\quad {when}}\quad \middle| x \middle| {\geq {\frac{1}{2}\quad {and}\quad {\gamma (x)}}} \right. = \left. {1\quad {when}}\quad \middle| x \middle| {< \frac{1}{2}} \right.}$

(4004): the algorithm iterates the loop on wlarg until a sufficientprecision is achieved. For example, it is possible to have$\lim = {\frac{\lambda}{8}{wpixels}}$

(4005): the value w_(f) displayed corresponds to w₀. We have: w₁=L−w_(f)and w₀=w_(f)

The value rapport displayed corresponds to:${rapport} = \frac{2{f\left( {{wf},\frac{wlarg}{wpixels}} \right)}}{\sum\limits_{{({{ni},{nj},{nk}})} \in {Es}}\left( \frac{\left| {{Fsmin}\left\lbrack {{ni},{nj},{nk}} \right\rbrack} \middle| {}_{2}{+ \left| {{Fsmax}\left\lbrack {{ni},{nj},{nk}} \right\rbrack} \right|^{2}} \right.}{\left| {{Btot}\left\lbrack {{ni},{nj},{nk}} \right\rbrack} \right|^{2}} \right)}$

The value rapport displayed characterizes the quality of the cross checkobtained between the images calculated from the frequencyrepresentations Famin and Famax. The closer it is to 1, the better thiscross check. When the sample is outside the observation region, thisvalue approaches 0.

7.15.3. Focussing

The focussing adjustment consists in correctly positioning the samplewithin the region of observation of the objectives. During thisadjustment, the values of rapport and w_(f) must be recalculatedconstantly. The position of the sample must be adjusted along the axis(2263) so as to obtain a sufficiently high value of rapport, and then itcan be adjusted more finely to obtain, for example,$w_{f} = {\frac{L}{2}.}$

This adjustment generally allows a first focussing. However, if, forexample, the index n_(o) is close to n_(v), this focussing is veryimprecise.

In every case, this adjustment must subsequently be completed by a moreprecise focus on the region of interest, as indicated in 7.17.3.

7.16. Obtaining the aberration compensation function

The function $^{{- j}\quad 2\quad \pi \frac{\Delta}{\lambda}}$

allowing, in principle, the correction of the phase differencesintroduced by the object and corresponding to spherical aberrationcomprises high frequencies which are filtered by the diaphragm. It canthus not be used directly and it is necessary to filter it to obtain, inthe form of an array of dimensions N_(pix)×N_(pix), a utilizablecorrection function.

The aberration compensation function, which will be used in the imagingphase, is obtained as follows:

Step 1: generation of arrays Ds_(p) of dimensions N_(e)×N_(e) with forexample N_(e)=4096:${{Ds}_{p}\left\lbrack {i,j} \right\rbrack} = {\exp \left\lbrack {{- \overset{\sim}{j}}\quad \frac{2\pi}{\lambda}\left( {{n_{0}w_{p}\sqrt{1 - {\left( \frac{n_{v}}{n_{0}} \right)^{2}\quad \frac{{ic}^{2} + {jc}^{2}}{r^{2}K_{p}^{2}}}}} - {{n_{v}\left( {w_{p} - \frac{z}{2}} \right)}\sqrt{1 - \frac{{ic}^{2} + {jc}^{2}}{r^{2}K_{p}^{2}}}} + {n_{v}\left( {{\frac{x}{2}\frac{ic}{{rK}_{p}}} + {\frac{y}{2}\frac{jc}{{rK}_{p}}}} \right)}} \right)} \right\rbrack}$

 where${{ic} = {i - \frac{N_{e}}{2}}},{{jc} = {j - \frac{N_{e}}{2}}},{r = \frac{N_{e}}{N_{pix}}}$

 the array Ds_(p) corresponds to the function$^{{- j}\quad 2\quad \pi \frac{\Delta}{\lambda}}$

 oversampled with a sufficiently short step to prevent aliasing.

Step 2: inverse Fourier transformation of arrays Ds_(p)

Step 3: extraction of middle part of the array, of dimensionsN_(pix)×N_(pix) with simulation of the diaphragm.

The program carries out:${D_{p}\left\lbrack {i,j} \right\rbrack} = {{Ds}_{p}\left\lbrack {{i - \frac{N_{pix}}{2} + \frac{N_{c}}{2}},{j - \frac{N_{pix}}{2} + \frac{N_{c}}{2}}} \right\rbrack}$

for all the pairs (i,j) such that${\left( {i - \frac{N_{pix}}{2}} \right)^{2} + \left( {j - \frac{N_{pix}}{2}} \right)^{2}} \leq \left( \frac{N_{pix}}{2} \right)^{2}$

where ouv is the aperture of the objectives.

Step 4: Fourier transformation of array D_(p).

One thus obtains, in the form of the array D_(p), the utilizablecorrection function.

7.17. Obtaining three-dimensional images

7.17.1. Principles

7.17.1.1. Superposition of the frequency representations

We saw that, for a given illuminating beam (index k,p), one obtains twotwo-dimensional images corresponding to the two sensors and referencedby the index q. When the point of direct impact goes through the path ofFIG. 56 on the sensor number 0, it is possible to generate a frequencyrepresentation from the two-dimensional images obtained on the twosensors. FIG. 58 shows how a set of two-dimensional frequencyrepresentations generates a three-dimensional representation. Atwo-dimensional representation is composed of a sphere portion (4101)obtained on sensor number 0 and a sphere portion (4102) obtained onsensor number 1. When the point of direct impact moves over a transversepath (3904), the movement of these sphere portions generates a volume.In FIG. 58 is represented a set of such sphere portions obtained forvarious positions of the point of direct impact on a transverse path.When the point of direct impact moves on the circle (3903), the volume(4104) delimited by (4105) is generated in addition. When the point ofdirect impact is on sensor number 1, a symmetrical volume is generated.

Four partial three-dimensional frequency representations aredistinguished which will be denoted F_(p,q) where the pair (p,q)designates a pair (sensor receiving the direct illuminating wave, sensorfrom which the two-dimensional representations allowing the generationof F_(p,q) result) with p=0 for sensor (2239), p=1 for (2229), q=0 whenit designates the same sensor as p, and q=1 when it designates theopposite sensor. The final three-dimensional representation is obtainedby superposing these partial three-dimensional representations.

The complete frequency representation obtained is represented in sectionin FIG. 59. It is composed:

of a part (4111) obtained by sensor 0 or 1 receiving the point of directimpact of the beam, and corresponding to the representations F_(0,0) andF_(1,0) which occupy the same part of the frequency space.

of a part (4113) obtained by sensor 1 when the point of direct impact ofthe beam is on sensor 0, and hence corresponding to the representationF_(0,1).

of a part (4112) obtained by sensor 0 when the point of direct impact ofthe beam is on sensor 1, and hence corresponding to the representationF_(1,1).

To obtain this volume exactly, it is necessary in principle to gothrough all the possible frequency values, i.e. N_(pix)×N_(pix) valueson each sensor, less the values located outside the zone limited by theaperture of the objective. Nevertheless, using the reduced path of FIG.56, a volume little different from the one drawn is obtained.

The path of FIG. 56 however constitutes a simple example and varioustypes of path may be used in practice. A few examples are given below:

A circle as in the first embodiment.

The path of FIG. 56, which allows images of better definition to beobtained.

The path of FIG. 56 made less dense. It is possible, for example to useone pixel out of two along this path. This has the effect of limitingthe thickness of the samples that may be observed under good conditions.

A complete path, i.e. defined by the arrays Io[k] and Jo[k] such thatthe point of coordinates (Io[k],Jo[k]) passes through the entire disclimited by the aperture of the objective. This means that each pixelwithin the disc limited by the circle (3902) of FIG. 56 must be reachedonce and only once. This path affords only a small improvement indefinition compared with that of FIG. 56 and considerably increases theacquisition time. On the other hand, the normal utilization conditionsof this microscope require that the diffracted beam remain of smallintensity in relation to the illuminating beam. The use of a completepath makes the system more robust when these utilization conditions arenot complied with.

7.17.1.2. Phase and intensity reference

In order to be able to combine the two-dimensional frequencyrepresentations to obtain partial three-dimensional frequencyrepresentations, it is necessary to establish for them a common phaseand intensity reference.

The complex values of the waves received on the directly illuminatedsensor can be referred to the value of the illuminating wave at itspoint of direct impact, as in the first embodiment or in the calculationof w_(p).

A more elaborate method is required for waves received on the oppositesensor. The reference illuminations are used to establish a ratiocharacteristic of the phase variation on each sensor and this ratio istaken into account to cancel these variations before referring to thevalue of the illuminating wave at its point of direct impact. This makesit possible to make mutually coherent the two-dimensionalrepresentations corresponding to a given pair p,q, independent of thevibrations affecting the system, and hence to establish for each pairp,q a three-dimensional frequency representation.

However, this does not make it possible to establish the phaserelationship between each of these three-dimensional frequencyrepresentations, which must be known in order to combine them into asingle frequency representation. The phase relationship between, forexample, the three-dimensional frequency representations F_(0,0) andF_(0,1), which correspond respectively to the parts (4111) and (4113) ofthe three-dimensional representation of the whole, can be establishedwhen these two parts have a common part (4114). It is sufficient tochoose the phase difference which causes the best matching of these tworepresentations on their common part. In order for there to be a commonpart, the aperture of the objective must be sufficient. The conditionfor the existence of this common part is determined geometrically andis: $n_{0} < {\frac{3}{2\sqrt{2}}\quad {ouv}}$

where ouv is the aperture of the objective. For example, for an apertureof 1.4, one obtains a maximum index of n₀=1.48 for the sample.

When the two representations F_(0,0) and F_(0,1) do not match, it is inprinciple impossible to determine their phase relationship: in factevery phase relationship corresponds to a possible frequencyrepresentation. A representation of the object in absorptivity and indexcan however be obtained from F_(0,0) alone, or a representation of theobject not differentiating the index and absorptivity can be obtainedfrom F_(0,1) alone by extracting the modulus of the spatialrepresentation obtained by using only F_(0,1).

7.17.1.2. Phase difference between two noisy arrays

We consider two arrays with a dimension A[i] and B[i] in which i variesfrom 0 to N−1. The elements of A (or B) are each affected by anindependent gaussian noise whose standard deviation is contained in anarray GA [i] (or GB[i]). With these arrays A and B assumed equal, exceptfor a constant phase and intensity ratio, and noise, an attempt is madeto determine this phase and intensity ratio, which will be denoted x,and which would be equal, in the absence of noise, to$x_{({noiseless})} = {\frac{B\lbrack i\rbrack}{A\lbrack i\rbrack}.}$

The value sought for x is that which maximizes the quantity P(x|A,B),representing the probability of a value of x knowing the arrays A and B.Maximizing this quantity is equivalent to maximizing P(B|A,x). For agiven value of x, and in the case in which the values of B[i] and A[i]are sufficiently over the noise level, the law giving B[i] from A[i] isthe composite of two gaussian laws of respective standard deviations|x|GA[i] and GB[i]. It is thus a gaussian law of standard deviation{square root over (|x|²(GA[i])²+(GB[i])²)}. We thus have:$\left. {{P\quad \left( B \right.A},x} \right) = {\prod\limits_{i}^{\quad}\quad {\exp \quad \left\{ {- \frac{{{{B\lbrack i\rbrack} - {x\quad {A\lbrack i\rbrack}}}}^{2}}{{{x}^{2}\quad \left( {{GA}\lbrack i\rbrack} \right)^{2}} + \left( {{GB}\lbrack i\rbrack} \right)^{2}}} \right\}}}$

In the cases that will be of interest to us later |x| is always closeto 1. We thus have:$\left. {{P\quad \left( B \right.A},x} \right) \approx {\prod\limits_{i}^{\quad}\quad {\exp \quad \left\{ {- \frac{{{{B\lbrack i\rbrack} - {{xA}\lbrack i\rbrack}}}^{2}}{\quad {\left( {{GA}\lbrack i\rbrack} \right)^{2} + \left( {{GB}\lbrack i\rbrack} \right)^{2}}}} \right\}}}$

Maximizing this quantity is equivalent to minimizing the quantity:$\sum\limits_{i}^{\quad}\quad \frac{{{{B\lbrack i\rbrack} - {{xA}\lbrack i\rbrack}}}^{2}}{\quad {\left( {{GA}\lbrack i\rbrack} \right)^{2} + \left( {{GB}\lbrack i\rbrack} \right)^{2}}}$

which equals, after division by the factor$\frac{{{A\lbrack i\rbrack}}^{2}}{\quad {\left( {{GA}\lbrack i\rbrack} \right)^{2} + \left( {{GB}\lbrack i\rbrack} \right)^{2}}}$

independent of x:$\frac{\sum\limits_{i}^{\quad}\quad \frac{{{B\lbrack i\rbrack}}^{2}}{\quad {\left( {{GA}\lbrack i\rbrack} \right)^{2} + \left( {{GB}\lbrack i\rbrack} \right)^{2}}}}{\sum\limits_{i}^{\quad}\quad \frac{{{A\lbrack i\rbrack}}^{2}}{\quad {\left( {{GA}\lbrack i\rbrack} \right)^{2} + \left( {{GB}\lbrack i\rbrack} \right)^{2}}}} - {x\quad \frac{\sum\limits_{i}^{\quad}\quad \frac{{A\lbrack i\rbrack}\quad \overset{\_}{B\lbrack i\rbrack}}{\quad {\left( {{GA}\lbrack i\rbrack} \right)^{2} + \left( {{GB}\lbrack i\rbrack} \right)^{2}}}}{\sum\limits_{i}^{\quad}\quad \frac{{{A\lbrack i\rbrack}}^{2}}{\quad {\left( {{GA}\lbrack i\rbrack} \right)^{2} + \left( {{GB}\lbrack i\rbrack} \right)^{2}}}}} - {\overset{\_}{x}\frac{\sum\limits_{i}^{\quad}\quad \frac{\overset{\_}{A\lbrack i\rbrack}\quad {B\lbrack i\rbrack}}{\quad {\left( {{GA}\lbrack i\rbrack} \right)^{2} + \left( {{GB}\lbrack i\rbrack} \right)^{2}}}}{\sum\limits_{i}^{\quad}\quad \frac{{{A\lbrack i\rbrack}}^{2}}{\quad {\left( {{GA}\lbrack i\rbrack} \right)^{2} + \left( {{GB}\lbrack i\rbrack} \right)^{2}}}}} + {x}^{2}$

We verify that this quantity is equal to:${{x - \frac{\sum\limits_{i}^{\quad}\quad \frac{\overset{\_}{A\lbrack i\rbrack}\quad {B\lbrack i\rbrack}}{\quad {\left( {{GA}\lbrack i\rbrack} \right)^{2} + \left( {{GB}\lbrack i\rbrack} \right)^{2}}}}{\sum\limits_{i}^{\quad}\quad \frac{{{A\lbrack i\rbrack}}^{2}}{\quad {\left( {{GA}\lbrack i\rbrack} \right)^{2} + \left( {{GB}\lbrack i\rbrack} \right)^{2}}}}}}^{2} + \frac{\sum\limits_{i}^{\quad}\quad \frac{{{B\lbrack i\rbrack}}^{2}}{\quad {\left( {{GA}\lbrack i\rbrack} \right)^{2} + \left( {{GB}\lbrack i\rbrack} \right)^{2}}}}{\sum\limits_{i}^{\quad}\quad \frac{{{A\lbrack i\rbrack}}^{2}}{\quad {\left( {{GA}\lbrack i\rbrack} \right)^{2} + \left( {{GB}\lbrack i\rbrack} \right)^{2}}}} - {\frac{\sum\limits_{i}^{\quad}\quad \frac{\overset{\_}{A\lbrack i\rbrack}\quad {B\lbrack i\rbrack}}{\quad {\left( {{GA}\lbrack i\rbrack} \right)^{2} + \left( {{GB}\lbrack i\rbrack} \right)^{2}}}}{\sum\limits_{i}^{\quad}\quad \frac{{{A\lbrack i\rbrack}}^{2}}{\quad {\left( {{GA}\lbrack i\rbrack} \right)^{2} + \left( {{GB}\lbrack i\rbrack} \right)^{2}}}}}^{2}$

The solution minimizing this quantity is thus:$x = \frac{\sum\limits_{i}^{\quad}\quad \frac{\overset{\_}{A\lbrack i\rbrack}\quad {B\lbrack i\rbrack}}{\quad {\left( {{GA}\lbrack i\rbrack} \right)^{2} + \left( {{GB}\lbrack i\rbrack} \right)^{2}}}}{\sum\limits_{i}^{\quad}\quad \frac{{{A\lbrack i\rbrack}}^{2}}{\quad {\left( {{GA}\lbrack i\rbrack} \right)^{2} + \left( {{GB}\lbrack i\rbrack} \right)^{2}}}}$

This simple formula is however valid only if the values of B[i] and A[i]are sufficiently over the noise level. One way of avoiding values notcomplying with this condition is to limit the summing as follows:$x = \frac{\sum\limits_{i \in E}^{\quad}\quad \frac{\overset{\_}{A\lbrack i\rbrack}\quad {B\lbrack i\rbrack}}{\quad {\left( {{GA}\lbrack i\rbrack} \right)^{2} + \left( {{GB}\lbrack i\rbrack} \right)^{2}}}}{\sum\limits_{i \in E}^{\quad}\quad \frac{{{A\lbrack i\rbrack}}^{2}}{\quad {\left( {{GA}\lbrack i\rbrack} \right)^{2} + \left( {{GB}\lbrack i\rbrack} \right)^{2}}}}$

where$E = \left\{ {i\left. {\frac{{\overset{\_}{A\lbrack i\rbrack}\quad {B\lbrack i\rbrack}}}{\quad {\left( {{GA}\lbrack i\rbrack} \right)^{2} + \left( {{GB}\lbrack i\rbrack} \right)^{2}}} \geq {{Coef} \cdot {\max\limits_{j}\quad \left( \frac{{\overset{\_}{A\lbrack j\rbrack}\quad {B\lbrack j\rbrack}}}{\quad {\left( {{GA}\lbrack j\rbrack} \right)^{2} + \left( {{GB}\lbrack j\rbrack} \right)^{2}}} \right)}}} \right\}} \right.$

i.e. the sums are limited to the set E of the values of i such that$\frac{{\overset{\_}{A\lbrack i\rbrack}\quad {B\lbrack i\rbrack}}}{\quad {\left( {{GA}\lbrack i\rbrack} \right)^{2} + \left( {{GB}\lbrack i\rbrack} \right)^{2}}}$

is higher than the product of its maximum value by a coefficient Coefwhich can be, for example, equal to 0.5.

This method will be used subsequently, generalized to 2 or 3 dimensions,to determine the standard deviation between two partially matchingfrequency representations.

7.17.1.3. Combining a series of noisy elements

Let us consider an array with one dimension A[i] in which i varies from0 to N−1. The elements of A are each affected by an independent gaussiannoise whose standard deviation is contained in an array GA[i]. In theabsence of noise, the elements of A are all equal to a value x to bedetermined. x is the value which maximizes P(x|A). Maximizing thisquantity is equivalent to maximizing P(A|x). And we have:$\left. {P\quad {\left( A \right.}x} \right) = {\prod\limits_{i}\left\{ {- \frac{{{x - {A\lbrack i\rbrack}}}^{2}}{\left( {{GA}\lbrack i\rbrack} \right)^{2}}} \right\}}$

Maximizing this quantity is equivalent to minimizing the followingquantity:$\sum\limits_{i}\quad \frac{{{x - {A\lbrack i\rbrack}}}^{2}}{\left( {{GA}\lbrack i\rbrack} \right)^{2}}$

which equals, after division by the quantity$\sum\limits_{i}\quad \frac{1}{\left( {{GA}\lbrack i\rbrack} \right)^{2}}$

independent of x:$\frac{\sum\limits_{i}\quad \frac{{{A\lbrack i\rbrack}}^{2}}{\left( {{GA}\lbrack i\rbrack} \right)^{2}}}{\sum\limits_{i}\quad \frac{1}{\left( {{GA}\lbrack i\rbrack} \right)^{2}}} - {x\quad \frac{\sum\limits_{i}\frac{\overset{\_}{A\lbrack i\rbrack}}{\left( {{GA}\lbrack i\rbrack} \right)^{2}}}{\sum\limits_{i}\quad \frac{1}{\left( {{GA}\lbrack i\rbrack} \right)^{2}}}} - {\overset{\_}{x}\quad \frac{\sum\limits_{i}\quad \frac{A\lbrack i\rbrack}{\left( {{GA}\lbrack i\rbrack} \right)^{2}}}{\sum\limits_{i}\quad \frac{1}{\left( {{GA}\lbrack i\rbrack} \right)^{2}}}} + {x}^{2}$

We verify that this quantity is equal to:${{x - \frac{\sum\limits_{i}\quad \frac{A\lbrack i\rbrack}{\left( {{GA}\lbrack i\rbrack} \right)^{2}}}{\sum\limits_{i}\quad \frac{1}{\left( {{GA}\lbrack i\rbrack} \right)^{2}}}}}^{2} + \frac{\sum\limits_{i}\quad \frac{{{A\lbrack i\rbrack}}^{2}}{\left( {{GA}\lbrack i\rbrack} \right)^{2}}}{\sum\limits_{i}\quad \frac{1}{\left( {{GA}\lbrack i\rbrack} \right)^{2}}} - {\frac{\sum\limits_{i}\quad \frac{A\lbrack i\rbrack}{\left( {{GA}\lbrack i\rbrack} \right)^{2}}}{\sum\limits_{i}\quad \frac{1}{\left( {{GA}\lbrack i\rbrack} \right)^{2}}}}^{2}$

The solution x which minimizes it is thus:$x = \frac{\sum\limits_{i}\quad \frac{A\lbrack i\rbrack}{\left( {{GA}\lbrack i\rbrack} \right)^{2}}}{\sum\limits_{i}\quad \frac{1}{\left( {{GA}\lbrack i\rbrack} \right)^{2}}}$

The noise on x is then given by the addition in quadratic value of thenoises on each A[i]:$\sigma_{x}^{2} = \frac{\sum\limits_{i}\quad \frac{\left( {{GA}\lbrack i\rbrack} \right)^{2}}{\left( {{GA}\lbrack i\rbrack} \right)^{4}}}{\quad \left( {\sum\limits_{i}\quad \frac{1}{\left( {{GA}\lbrack i\rbrack} \right)^{2}}} \right)^{2}}$

We verify that this is equivalent to:$\frac{1}{\sigma_{x}^{2}} = {\sum\limits_{i}\quad \frac{1}{\left( {{GA}\lbrack i\rbrack} \right)^{2}}}$

This result will be used to determine a frequency representation fromseveral partial representations matching on certain points.

7.17.2 Algorithm

A series of images is first acquired as indicated in 7.12., the point ofdirect impact of the illuminating beam taking, on each sensor, a pathdefined as indicated in 7.17.1.1. For simplification, the acquisitionphase and the calculation phase are separated. However, thethree-dimensional frequency representations F_(p,q) can also begenerated as the acquisition progresses. For example, if a complete pathis used, the separation of the acquisition and calculation phasesrequires more memory and it is thus preferable to cany out thecalculation as the acquisition progresses.

The acquisition phase generates the following arrays:

M_(k,p,q)[i,j], corresponding to the main imaging

B_(k,p,q)[i,j] noise indicator

H_(k,p,q)[i,j] reference imaging for the k-th acquisition

BH_(k,p,q)[i,j] reference image noise indicator

where k indexes the image acquisition, p indexes the sensor illuminatedby the direct beam, q indicates:

q=0: sensor illuminated by the direct beam

q=1: opposite sensor.

The series of control indices used being defined by the arrays Ic andJc, the program also generates the series of coordinates of the pointsof direct and opposite impact of the illuminating beams.

imax_(k,p,q)=Ia[p,q,Ic[k,p],Jc[k,p]]

jmax_(k,p,q)=Ja[p,q,Ic[k,p],Jc[k,p]]

From these data, a program generates a three-dimensional spatialrepresentation of the studied object. This program comprises thefollowing steps:

Step 1. This step consists in calculating the ratio characteristic ofthe phase and amplitude difference due to vibrations and to fluctuationsin the intensity of the laser between a reference imaging operation ofindex k=0 and the constant reference imaging of index k. This differenceis characterized by variations in the function H_(k,p,q)[i,j] whichshould be constant in the absence of vibrations. For all the triplets(k,p,q), the program thus calculates, in accordance with the method seenin 7.17.2., the ratio:$R_{k,p,q} = \frac{\sum\limits_{{({i,j})} \in E_{k,p,q}}\quad \frac{{H_{0,p,q}\left\lbrack {i,j} \right\rbrack}\overset{\_}{H_{k,p,q}\left\lbrack {i,j} \right\rbrack}}{{{{BH}_{k,p,q}\left\lbrack {i,j} \right\rbrack}}^{2} + {{{BH}_{0,p,q}\left\lbrack {i,j} \right\rbrack}}^{2}}}{\sum\limits_{{({i,j})} \in E_{k,p,q}}\quad \frac{{{H_{k,p,q}\left\lbrack {i,j} \right\rbrack}}^{2}}{{{{BH}_{k,p,q}\left\lbrack {i,j} \right\rbrack}}^{2} + {{{BH}_{0,p,q}\left\lbrack {i,j} \right\rbrack}}^{2}}}$

 with:$\left. {E_{k,p,q} = \left\{ {\left( {i,j} \right){{\frac{{{H_{0,p,q}\left\lbrack {i,j} \right\rbrack}\overset{\_}{H_{k,p,q}\left\lbrack {i,j} \right\rbrack}}}{{{{BH}_{k,p,q}\left\lbrack {i,j} \right\rbrack}}^{2} + {{{BH}_{0,p,q}\left\lbrack {i,j} \right\rbrack}}^{2}} \geq {{{Coef} \cdot \underset{\underset{0 \leq b \leq N_{pix}^{- 1}}{0 \leq a \leq N_{pix}^{- 1}}}{\max.}}\quad \left( \frac{{{H_{0,p,q}\left\lbrack {a,b} \right\rbrack}\overset{\_}{H_{k,p,q}\left\lbrack {a,b} \right\rbrack}}}{{{{BH}_{k,p,q}\left\lbrack {a,b} \right\rbrack}}^{2} + {{{BH}_{0,p,q}\left\lbrack {a,b} \right\rbrack}}^{2}} \right)}}}} \right.} \right\}$

and for example Coef=0.5.

Step 2: This step consists in performing the operation consisting in:

normalizing each two-dimensional representation to compensate forvariations in the phase and amplitude differences due to vibrations,characterized by the quantity R_(k,p,q)

compensating for spherical aberration and the poor relative positioningof the objectives, characterized by

D _(p{overscore (q)}+{overscore (p)}q) [i,j]

going to the value of the illuminating wave at its point of directimpact.

The program thus goes through the indices k,p,q,i,j performing:${M_{k,p,q}\left\lbrack {i,j} \right\rbrack} = \frac{{M_{k,p,q}\left\lbrack {i,j} \right\rbrack}\quad {D_{{p\overset{\_}{q}} + {\overset{\_}{p}q}}\left\lbrack {i,j} \right\rbrack}\quad R_{k,p,q}}{{M_{k,p,0}\left\lbrack {{imax}_{k,p,0},{jmax}_{k,p,0}} \right\rbrack}\quad {D_{p}\left\lbrack {{imax}_{k,p,0},{jmax}_{k,p,0}} \right\rbrack}\quad R_{k,p,0}}$${B_{k,p,q}\left\lbrack {i,j} \right\rbrack} = {{B_{k,p,q}\left\lbrack {i,j} \right\rbrack}{\frac{{D_{{p\overset{\_}{q}} + {\overset{\_}{p}q}}\left\lbrack {i,j} \right\rbrack}\quad R_{k,p,q}}{{M_{k,p,0}\left\lbrack {{imax}_{k,p,0},{jmax}_{k,p,0}} \right\rbrack}\quad {D_{p}\left\lbrack {{imax}_{k,p,0},{jmax}_{k,p,0}} \right\rbrack}\quad R_{k,p,0}}}}$

The use of the array D_(p), which is the result of steps 7.11, 7.15. and7.16., makes it possible to clearly improve the results when the averageindex of the object differs from the nominal index of the objectives. Itis however also possible to omit the steps 7.11, 7.15. and 7.16. Theposition adjustment of the objectives described in 7.10 must then becarried out so as to obtain a centered and punctual spatial image. Thearray D_(p) must then be set to 1.

The use of the values R_(k,p,q) enables compensation for possiblevibrations in the optical table. However, if the optical table isperfectly stable, this compensation is not necessary. The valuesR_(k,p,q) must then be set to 1.

Step 3: This step consists in calculating for each pair (p,q) athree-dimensional frequency representation F_(p,q), combined with anarray IB_(p,q) of real values, containing the inverse of the square ofthe standard deviation of the gaussian noise affecting each element ofthe array F_(p,q). These arrays have the dimensions 2 N_(pix)×2N_(pix)×2 N_(pix). Each point of a two-dimensional frequencyrepresentation corresponds to a point of the three-dimensional frequencyrepresentation F_(p,q), whose coordinates must be determined. When apoint is obtained several times, the most probable value is determined.

The program initializes the arrays F_(p,q), IB_(p,q) to zero and thengoes through all the indices p,q,k,i,j performing, for each quintuplet(p,q,k,i,j), the following operations numbered from 1 to 3:

Operation 1: calculation of the indices of the three-dimensionalfrequency representation. The program carries out:

ni=a _(p)(i−imax_(k,p,q))+N _(pix)

nj=a _(p)(j−jmax_(k,p,q))+N _(pix)

${nk} = {\sqrt{K_{m}^{2} - \left( {a_{p}^{2}\left( {i - \frac{N_{pix}}{2}} \right)} \right)^{2} - \left( {a_{p}^{2}\left( {j - \frac{N_{pix}}{2}} \right)} \right)^{2}} - \sqrt{K_{m}^{2} - \left( {a_{p}^{2}\left( {{imax}_{k,p,q} - \frac{N_{pix}}{2}} \right)} \right)^{2} - \left( {a_{p}^{2}\left( {{jmax}_{k,p,q} - \frac{N_{pix}}{2}} \right)} \right)^{2}} + N_{pix}}$

A distance of one pixel, measured on the sensor p, corresponds to a realfrequency deviation proportional to $\frac{1}{K_{p}}.$

The pixels consequently do not represent the same frequency deviationson the two sensors. One obtains a common unit proportional to thefrequency deviations by multiplying the distances obtained on the sensorp by the coefficient $a_{p} = {\frac{K_{o} + K_{1}}{2\quad K_{p}}.}$

A value of K then becomes common to the two sensors and equals$\frac{K_{0} + K_{1}}{2}.$

It must be corrected to take into account the index of the sample andone thus obtains$K_{m} = {\frac{n_{o}}{n_{v}}\quad {\frac{K_{0} + K_{1}}{2}.}}$

Operation 2: modification of the indices of the three-dimensionalfrequency representation in the case of q=1.

If q=1, the frequency corresponding to the coordinatesimax_(k,p,q),jmax_(k,p,q) is not the zero frequency.

In fact, one has, at this point, by resuming the notations used in 5.3.:f_(c)=−f_(e) and hence f_(t)=f_(c)−f_(e)=−2f_(e). The frequency obtainedusing the preceding method must thus be translated by a vector −2f_(e),thus resulting in the following additional operations carried out onlyin the case of q=1:

ni+=2a _(p) imax_(k,p,1)

nj+=2a _(p) jmax_(k,p,1)

${nk}\text{+} = 2\sqrt{K_{m}^{2} - {a_{p}^{2}\left( {{imax}_{k,p,1} - \frac{N_{pix}}{2}} \right)}^{2} - {a_{p}^{2}\left( {{jmax}_{k,p,1} - \frac{N_{pix}}{2}} \right)}^{2}}$

 With the calculation of the indices ni,nj,nk leading to non-integervalues, the integer closest to the calculated value is assigned to them.

Operation 3: modification of array elements.

Having generated the modified indices, the program modifies the arrayelements:${{{IB}_{p,q}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}\text{+}} = \frac{1}{\left| {B_{k,p,q}\left\lbrack {i,j} \right\rbrack} \right|^{2}}$${{F_{p,q}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}\text{+}} = \frac{M_{k,p,q}\left\lbrack {i,j} \right\rbrack}{\left| {B_{k,p,q}\left\lbrack {i,j} \right\rbrack} \right|^{2}}$

Step 4: An operation remains to be carried out in order to obtain themost probable value on each frequency. The program thus goes through theindices p,q,ni,nj,nk performing, whenever IB_(p,q)[ni,nj,nk]≠0, theoperation:${F_{p,q}\left\lbrack {{ni},{nj},{nk}} \right\rbrack} = \frac{F_{p,q}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}{{IB}_{p,q}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}$

Step 5: The coordinate systems in which the indices i,j and henceni,nj,nk have been evaluated are inverted between the two sensors. It isthus necessary to make a change of coordinate system for representationscorresponding to the sensor indexed 1 to express them in the samecoordinate system as the representations corresponding to the sensorindexed 0. The program consequently carries out the change of variablesni→2N_(pix)−ni, nj→2N_(pix)−nj, nk→2N_(pix)−nk in the arrayscorresponding to the sensor indexed 1 in order to express all thefrequencies in the same coordinate system. The arrays corresponding to afrequency representation coming from sensor 1 have an index p equal to 1or 0 and an index q equal to {overscore (p)}.

To make these changes of variable, the program goes through all theindices p,ni,nj,nk, performing:

F _(p,{overscore (p)}) [ni,nj,nk]=F _(p,{overscore (p)})[2N _(pix)−ni,2N _(pix) −nj,2N _(pix) −nk]

IB _(p,{overscore (p)}) [ni,nj,nk]=IB _(p,{overscore (p)})[2N _(pix)−ni,2N _(pix) −nj,2N _(pix) −nk]

Step 6: The program calculates the characteristic ratio of the phase andamplitude difference between the wave received on the sensor illuminateddirectly and the one received on the sensor not illuminated. It thusgoes through the indices p=0, p=1, performing, in accordance with theprinciple seen in 7.17.1.2.:${Rb}_{p} = \frac{\sum\limits_{{({{ni},{nj},{nk}})} \in E_{p}}\left\lbrack \frac{{F_{p,0}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}\quad \overset{\_}{F_{p,1}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}}{\frac{1}{{IB}_{p,0}\left\lbrack {{ni},{nj},{nk}} \right\rbrack} + \frac{1}{{IB}_{p,1}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}} \right\rbrack}{\sum\limits_{{({{ni},{nj},{nk}})} \in E_{p}}\left\lbrack \frac{\left| {F_{p,1}\left\lbrack {{ni},{nj},{nk}} \right\rbrack} \right|^{2}}{\frac{1}{{IB}_{p,0}\left\lbrack {{ni},{nj},{nk}} \right\rbrack} + \frac{1}{{IB}_{p,1}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}} \right\rbrack}$

In this expression, the sums are restricted to a set E_(p) made up oftriplets (ni,nj,nk) complying withIB_(p,0)[ni,nj,nk]IB_(p,1)[ni,nj,nk]≠0 and$\frac{\left| {{F_{p,0}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}\quad \overset{\_}{F_{p,1}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}} \right|}{\frac{1}{{IB}_{p,0}\left\lbrack {{ni},{nj},{nk}} \right\rbrack} + \frac{1}{{IB}_{p,1}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}} \geq {{Coef} \cdot \quad {\max\limits_{\underset{\underset{0 \leq c \leq {{2\quad N_{pix}} - 1}}{0 \leq b \leq {{2\quad N_{pix}} - 1}}}{0 \leq a \leq {{2\quad N_{pix}} - 1}}}\left( \frac{\left| {{F_{p,0}\left\lbrack {a,b,c} \right\rbrack}\quad \overset{\_}{F_{p,1}\left\lbrack {a,b,c} \right\rbrack}} \right|}{\frac{1}{{IB}_{p,0}\left\lbrack {a,b,c} \right\rbrack} + \frac{1}{{IB}_{p,1}\left\lbrack {a,b,c} \right\rbrack}} \right)}}$

where, for example, Coef=0,5

Step 7: The program modifies the three-dimensional representationsobtained from sensors not illuminated directly. It goes through theindices p, ni,nj,nk, performing:

F _(p,1) [ni,nj,nk]=F _(p,1) [ni,nj,nk]Rb _(p)

Step 8: The program calculates the final frequency representationcontained in an array F of dimensions 2N_(pix)×2N_(pix)×2N_(pix). Itinitializes this array to 0 and then goes through the indices ni,nj,nkwhile testing the condition:${\sum\limits_{p,q}{{IB}_{p,q}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}} \neq 0$

 When the condition is met, it performs:${F\left\lbrack {{ni},{nj},{nk}} \right\rbrack} = \frac{\sum\limits_{p,q}{{F_{p,q}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}\quad {{IB}_{p,q}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}}}{\sum\limits_{p,q}{{IB}_{p,q}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}}$

Step 9: The program carries out a three-dimensional inverse Fouriertransformation of the frequency representation thus obtained to obtain aspatial representation.

Step 10: As in the first embodiment, the program can then display therepresentation thus obtained in the form of sections or projectionswhich may be stereoscopic.

7.17.4. Focussing

The algorithm described 7.17.2. makes it possible to obtainthree-dimensional representations of the sample. The focussingadjustment consists in adjusting the position of the object so thatthese representations are those of a region of interest of the sample.This can be accomplished by the operator, moving the sample whileobserving, for example, a plane projection or a section of thisthree-dimensional representation, moving the object to obtain an imageof the region of interest. If the sample is moved in the direction ofthe optical axis, this modifies the values of w_(p), and the proceduredescribed in 7.15.2. must be reiterated to obtain a correct value ofw_(p).

7.18. Variants

The algorithms used in the present embodiment allow many variants, someof which are set forth below.

7.18.1. Use of pre-recorded values of the direct illuminating beam

This variant consists in modifying Step 2 of the algorithm described in7.17.2. so as to use the pre-recorded values of the direct illumination.One in fact has, to within a phase factor which is constant over all thevalues of k,p, imax_(k,p,0),jmax_(k,p,0):${M_{k,p,0}\left\lbrack {{imax}_{k,p,0},{jmax}_{k,p,0}} \right\rbrack} = \frac{{{Ra}\left\lbrack {p,{{Ic}\left\lbrack {k,p} \right\rbrack},{{Jc}\left\lbrack {k,p} \right\rbrack}} \right\rbrack}{{Rb}\left\lbrack {p,{{Ic}\left\lbrack {k,p} \right\rbrack},{{Jc}\left\lbrack {k,p} \right\rbrack}} \right\rbrack}}{R_{k,p,0}}$

This value may be introduced in the formula used in Step 2, which isthus replaced by:${M_{k,p,q}\left\lbrack {i,j} \right\rbrack} = \frac{\left( {{M_{k,p,q}\left\lbrack {i,j} \right\rbrack}\quad {D_{{p\overset{\_}{q}} + {\overset{\_}{p}q}}\left\lbrack {i,j} \right\rbrack}\quad R_{k,p,q}} \right)}{\left( {{{Ra}\left\lbrack {p,{{Ic}\left\lbrack {k,p} \right\rbrack},{{Jc}\left\lbrack {k,p} \right\rbrack}} \right\rbrack}\quad {{Rb}\left\lbrack {p,{{Ic}\left\lbrack {k,p} \right\rbrack},{{Jc}\left\lbrack {k,p} \right\rbrack}} \right\rbrack}\quad {D_{p}\left\lbrack {{imax}_{k,p,0},{jmax}_{k,p,0}} \right\rbrack}} \right)}$

7.18.2. Use of precalculated values of the direct illuminating beam

Rb[p,Ic[k,p],Jc[k,p]] is in principle equal to the function obtained in7.7.:$^{\overset{\_}{j}\quad 2\quad \pi \quad \frac{\Delta}{\lambda}} = {\exp \left\{ {\overset{\sim}{j}\quad {\frac{2\quad \pi}{\lambda}\left\lbrack {{n_{v}\quad \left( {{x\quad \frac{i}{K_{p}}} + {y\quad \frac{j}{K_{p}}} + {z\quad \frac{1}{K_{p}}\quad \sqrt{K_{p}^{2} - i^{2} - j^{2}}}} \right)} + {L\quad \left( {{n_{o}\quad \sqrt{1 - {\left( \frac{n_{v}}{n_{o}} \right)^{2}\quad \frac{i^{2} + j^{2}}{K_{p}^{2}}}}} - {n_{v}\quad \sqrt{1 - \frac{i^{2} + j^{2}}{K_{p}^{2}}}}} \right)}} \right\rbrack}} \right\}}$

this function being however filtered by the diaphragm. It is thuspossible to replace Rb[p,Ic[k,p],Jc[k,p]] byRB_(p)[imax_(k,p,0),jmax_(k,p,0)] in which the function RB_(p)[i,j] isobtained as follows (this method is similar to the one used forD_(p)[i,j] in 7.16.)

Step 1: generation of arrays RB_(p) of dimensions N_(e)×N_(e) with, forexample, N_(e)=4096:${{RB}_{p}\left\lbrack {i,j} \right\rbrack} = {\exp \left\{ {\overset{\sim}{j}\quad {\frac{2\quad \pi}{\lambda}\left\lbrack {{n_{v}\quad \left( {{x\quad \frac{ic}{K_{p}}} + {y\quad \frac{jc}{K_{p}}} + {z\quad \frac{1}{K_{p}}\quad \sqrt{K_{p}^{2} - {ic}^{2} - {jc}^{2}}}} \right)} + {L\quad \left( {{n_{o}\quad \sqrt{1 - {\left( \frac{n_{v}}{n_{o}} \right)^{2}\quad \frac{{ic}^{2} + {jc}^{2}}{K_{p}^{2}}}}} - {n_{v}\quad \sqrt{1 - \frac{{ic}^{2} + {jc}^{2}}{K_{p}^{2}}}}} \right)}} \right\rbrack}} \right\}}$

 where${{ic} = {i - \frac{N_{e}}{2}}},{{jc} = {j - \frac{N_{e}}{2}}},{r = \frac{N_{e}}{N_{pix}}}$

Step 2: inverse Fourier transformation of arrays RB_(p)

Step 3: extraction of middle part of the array, of dimensionsN_(pix)×N_(pix) with simulation of the diaphragm.

The program carries out:${{RB}_{p}\left\lbrack {i,j} \right\rbrack} = {{RB}_{p}\left\lbrack {{i - \frac{N_{pix}}{2} + \frac{N_{c}}{2}},{j - \frac{N_{pix}}{2} + \frac{N_{c}}{2}}} \right\rbrack}$

 for all the pairs (i,j) such that${\left( {i - \frac{N_{pix}}{2}} \right)^{2} + \left( {j - \frac{N_{pix}}{2}} \right)^{2}} \leq \left( \frac{N_{pix}}{2} \right)^{2}$

Step 4: Fourier transformation of array RB_(p).

One then obtains in the form of the array RB_(p) the function equivalentto array Rb Step 2 of the algorithm described in 7.17.2. is thenreplaced by:${{M_{k,p,q}\left\lbrack {i,j} \right\rbrack} = \frac{\left( {{M_{k,p,q}\left\lbrack {i,j} \right\rbrack}\quad {D_{{p\overset{\_}{q}} + {\overset{\_}{p}q}}\left\lbrack {i,j} \right\rbrack}\quad R_{k,p,q}} \right)}{\left( {{{Ra}\left\lbrack {p,{{Ic}\left\lbrack {k,p} \right\rbrack},{{Jc}\left\lbrack {k,p} \right\rbrack}} \right\rbrack}\quad {{RB}_{p}\left\lbrack {{imax}_{k,p,0}{jmax}_{k,p,0}} \right\rbrack}\quad {D_{p}\left\lbrack {{imax}_{k,p,0},{jmax}_{k,p,0}} \right\rbrack}} \right)}}$

This replacement is similar to a “smoothing” of the function defined byarray Rb and can in certain cases improve the results, in particular ifthe diffracted wave is strong, falling outside the normal utilizationconditions of this microscope. In this case, this formula will becombined with the use of a complete path as defined in 7.17.1.1.

7.18.3. Obtaining confocal representations

A confocal microscope makes it possible to obtain three-dimensionalspatial representations, which will be called confocal representations.The present microscope allows a confocal representation to be obtainedwhich is strictly equivalent to that which would be obtained by means ofa confocal microscope.

In fact, the illuminating wave used by a confocal microscope is the sumof the plane waves used in the case in which a complete path is used forthe acquisition; each plane wave has to be assigned a phase dependent onthe illuminated point. The wave equivalent to the wave received by aconfocal microscope of the same aperture as the present microscope, whenthe central point is illuminated, can thus be generated by summing thetwo-dimensional representations of diffracted waves obtained for all theilluminating waves forming a complete path.

It can be demonstrated that the confocal representation of the object isthe inverse Fourier transform of a three-dimensional frequencyrepresentation obtained by summing the partial two-dimensional frequencyrepresentations obtained from each illuminating wave, all theilluminating waves going through a complete path.

In addition, a confocal microscope acquires images only on a singleobjective and illuminates the sample only on one side. Furthermore, itgenerates a value which is the intensity of the wave having traversedthe object and not its complex value. The confocal representation inintensity is thus obtained from the waves received by a singleobjective, and by extracting the square of the modulus of the previouslyobtained confocal representation. Finally, the confocal microscope doesnot correct the spherical aberration due to the index of the object.

A confocal representation can thus be obtained by the present microscopeby using a complete path for the acquisition, as defined in 7.17.1.1.,and by modifying the procedure described in Paragraph 17.2. as follows:

(1) Step 2 is modified as follows:${{M_{k,p,q}\left\lbrack {i,j} \right\rbrack} = \frac{\left( {{M_{k,p,q}\left\lbrack {i,j} \right\rbrack}\quad {D_{{p\overset{\_}{q}} + {\overset{\_}{p}q}}\left\lbrack {i,j} \right\rbrack}\quad R_{k,p,q}} \right)}{\left( {{{Ra}\left\lbrack {p,{{Ic}\left\lbrack {k,p} \right\rbrack},{{Jc}\left\lbrack {k,p} \right\rbrack}} \right\rbrack}\quad {{RB}_{p}\left\lbrack {{imax}_{k,p,0}{jmax}_{k,p,0}} \right\rbrack}\quad {D_{p}\left\lbrack {{imax}_{k,p,0},{jmax}_{k,p,0}} \right\rbrack}} \right)}}$

 in which the arrays D_(p) can be set to 1 if no spherical aberration isto be corrected and in which RB_(p) is defined as in 7.17.3.2.

(2) Operation 3 of Step 3 is replaced byF_(p,q)[ni,nj,nk]+=M_(k,p,q)[i,j]

(3) Steps 4, 5, 6, 7 are not carried out.

(4) Step 8 is replaced by F[ni,nj,nk]=F_(po,qo)[ni,nj,nk] in which thechoice of the indices (po,qo) depends on the type of confocalrepresentation one seeks to generate.

if (po,qo)=(0,0) or (po,qo)=(1,0), a representation is generatedcorresponding to the one which would be obtained by a confocalmicroscope by reflection.

if (po,qo)=(0,1) or (po,qo)=(1,1), a representation is generatedcorresponding to the one which would be obtained by a confocalmicroscope by transmission.

(5) The square of the modulus of the spatial representation obtainedafter Step 9 then corresponds to the confocal representation inintensity.

Replacing the calculation of the most probable value of the frequencyrepresentation at each point results in an overevaluation of the lowfrequencies in relation to the high frequencies, which is equivalent toa filtering of high frequencies and hence to a loss of definition.

It is also possible to obtain a confocal representation from thethree-dimensional frequency representation obtained in accordance withParagraph 17 unmodified: in fact, the three-dimensional representationof the object constitutes the most complete possible informationobtainable with objectives of a given aperture and can be used tosimulate any type of image which could be generated from any type ofmicroscope using the same objective and the same wavelength.

However, the use a complete path makes the system more robust, inaccordance with what was stated in 7.17.1.1. If a confocalrepresentation is obtained using a path such as the one in FIG. 56, itwill be disturbed in the case in which a major part of the illuminatingwave is diffracted, and this more significantly than the confocalrepresentation obtained by means of a confocal microscope or by the useof a complete path. It can consequently not be considered to berigorously equivalent to that generated by a confocal microscope.

7.18.4. Producing three-dimensional images with reference wave control

In the previously described methods, the phase shift device (2205) iscontrolled to generate phase differences θ_(d) in the illuminating wavedependent on the index d in accordance with the array indicated in7.12.2.1.

To obtain the present variant, this discrete phase shift device must bereplaced by a device allowing a continuous phase shift. Such a devicemay be a liquid crystal device placed between two polarizers, marketedfor example by the company Newport. By modifying the path of the beam,this device can also be a piezoelectric mirror as in the firstembodiment.

The present variant consists, during the image acquisition described in7.17.2. and carried out as indicated in 7.12.2.1., in controlling thephase shift device so as to replace the phase shift θ_(d) by a phaseshift$\Theta_{d}^{\prime} = {\theta_{d} - {{Arg}\quad \left( \frac{{{Ra}\left\lbrack {p,{{Ic}\left\lbrack {k,p} \right\rbrack},{{Jc}\left\lbrack {k,p} \right\rbrack},{{Jc}\left\lbrack {k,p} \right\rbrack}} \right\rbrack}{{Rb}\left\lbrack {p,{{Ic}\left\lbrack {k,p} \right\rbrack},{{Jc}\left\lbrack {k,p} \right\rbrack}} \right\rbrack}}{R_{k,p,0}} \right)}}$

in which Arg designates the argument of a complex number. This allowsthe cancellation of the illuminating wave phase at its point of directimpact and obviates the compensation of this phase. During thecalculation phase described in 7.17.2., in Step 2, the formula used maybe replaced by:${M_{k,p,q}\left\lbrack {i,j} \right\rbrack} = \frac{{M_{k,p,q}\left\lbrack {i,j} \right\rbrack}\quad {D_{{p\overset{\_}{q}} + {\overset{\_}{p}q}}\left\lbrack {i,j} \right\rbrack}\quad R_{k,p,q}}{{D_{p}\left\lbrack {{imax}_{k,p,0},{jmax}_{k,p,0}} \right\rbrack}\quad R_{k,p,0}}$

This mode is equivalent to controlling the phase difference of theilluminating beams by the phase shift device instead of compensating forit by calculation after acquisition.

7.18.5. Obtaining frequency representations without calculating the wavereceived on the receiving surface

If the optical table is of sufficient quality to eliminate vibrations,the formula used in 7.18.4. becomes:${M_{k,p,q}\left\lbrack {i,j} \right\rbrack} = \frac{{{M_{k,p,q}\left\lbrack {i,j} \right\rbrack}\quad {D_{{p\overset{\_}{q}} + {\overset{\_}{p}q}}\left\lbrack {i,j} \right\rbrack}}\quad}{D_{p}\left\lbrack {{imax}_{k,p,0},{jmax}_{k,p,0}} \right\rbrack}$

To simplify the explanations, it may be assumed that the beam attenuatorand the polarization rotators are not used. We then verify that eachfrequency representation F_(p,q) obtained in the procedure described in7.17. can be expressed in the following form:${F_{p,q}\left\lbrack {{ni},{nj},{nk}} \right\rbrack} = {{\frac{1}{6}\quad \left( {{2{F_{p,q,0}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}} - {F_{p,q,1}\left\lbrack {{ni},{nj},{nk}} \right\rbrack} - {F_{p,q,2}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}} \right)} + {\overset{\sim}{j}\quad \frac{1}{2\sqrt{3}}\quad \left( {{F_{p,q,1}\left\lbrack {{ni},{nj},{nk}} \right\rbrack} - {F_{p,q,2}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}} \right)}}$

where F_(p,q,d)[ni,nj,nk] is obtained like F_(p,q) in procedure 7.17.unmodified, but by replacing M_(k,p,q)[i,j] by the real valueM0[k,p][0,d,0,0][q,i,j] obtained in the procedure described in 7.12. fora corresponding value of the index d indexing the phase shift. It isthus possible to calculate for each index d a separate frequencyrepresentation F_(p,q,d), these representations then being superimposedto obtain the frequency representation F_(p,q), instead of performingdirectly in the procedure 7.12. the superposition of the valuescorresponding to each index d.

It is also possible to carry out the superposition of the arrayscorresponding to each index d after passing into the spatial domain byinverse Fourier transformation.

Finally, it is possible not to use the same points of impact of theilluminating wave according to the phase shift applied. In this case,each phase shift corresponds to a distinct path and the arraysIo[k],Jo[k] must be replaced by the arrays Io[k,d],Jo[kd]. It is thenpossible, as previously, to calculate separate arrays F_(p,q,d) beforesuperposing them to obtain the arrays F_(p,q).

This calculation mode is not particularly advantageous but shows that itis not indispensable to calculate the complex two-dimensionalrepresentations in an intermediate phase, nor even to undertake anyacquisition of data corresponding to these complex two-dimensionalrepresentations.

7.18.6. Obtaining images with a single value of the phase shift

The present variant consists in modifying the procedure described in7.12. so as to acquire only the real part of the complex number normallyacquired as described in 7.12. This real part may be acquired in asingle step, thus allowing the use of a single value of the index dcharacterizing the phase shift. Since only the real part is acquired,the frequency representation obtained, assuming that the sphericalaberration compensation array D_(p) is set to 1, is the real part of thecomplex representation. The spatial representation obtained by inverseFourier transformation is then the superposition of the normal imagewith a conjugate image symmetrical in relation to the origin of thereference wave. In order for the normal image not to be superimposed onits symmetrical image, the origin of the reference wave must be placedon the side of the diaphragm, slightly outside the aperture of thediaphragm, and not at its center. The aperture of the diaphragm must bereduced by half in order to avoid the aliasing induced by thisdisplacement of the reference wave origin. The image finally obtainedthen comprises the normal image and the symmetrical image, notsuperimposed and thus utilizable. However, in order to be able toacquire the real part in a single step, the reference wave intensitymust be sufficiently higher than the diffracted wave intensity, so asnot to induce second-order errors. The quality of the image finallyobtained thus depends on the intensity of the reference wave. Too low anintensity induces second-order distortions and too high an intensityincreases the gaussian noise.

To best comply with the condition of sufficient reference waveintensity, the intensity of the reference wave alone must be adjustednot at one-fourth the maximum value of the digitizer as indicated in7.4. but, for example, at 80% of this value.

In order for the real part to be actually acquired with each imagingoperation, the single phase shift used must make it possible to obtaindirectly a constant phase reference. This shift will consequently besimilar to what was done in 7.18.4.:

Θ₀=−Arg(Ra[p,Ic[k,p],Jc[k,p]]Rb[p,Ic[k,p],Jc[k,p]])

Despite the application of this phase shift, the phase reference may notbe constant in the presence of optical table vibrations. This woulddestroy the image and it is thus necessary to use an optical table ofvery good quality to eliminate these vibrations.

The index d takes on a single value instead of three and Step 2 of theprocedure described in 7.12.2.2. is replaced by:${{{M2}\left\lbrack {r_{1},r_{2}} \right\rbrack}\quad\left\lbrack {q,i,j} \right\rbrack} = \frac{{{{{M0}\left\lbrack {k,p} \right\rbrack}\left\lbrack {{{M1}\left\lbrack {q,i,j} \right\rbrack},0,r_{1},r_{2}} \right\rbrack}\left\lbrack {q,i,j} \right\rbrack} - {{Iref}\left\lbrack {{{\overset{\_}{p}q} + {p\overset{\_}{q}}},i,j} \right\rbrack}}{{{att}\left\lbrack {{M1}\left\lbrack {q,i,j} \right\rbrack} \right\rbrack}\sqrt{{Iref}\left\lbrack {{{\overset{\_}{p}q} + {p\overset{\_}{q}}},i,j} \right\rbrack}}$

As in 7.18.4., but taking into account the absence of vibrations, Step 2of the procedure described in 7.17.2. is replaced by:${M_{k,p,q}\left\lbrack {i,j} \right\rbrack} = \frac{{M_{k,p,q}\left\lbrack {i,j} \right\rbrack}{D_{{p\overset{\_}{q}} + {\overset{\_}{p}q}}\left\lbrack {i,j} \right\rbrack}}{D_{p}\left\lbrack {{imax}_{k,p,0},{jmax}_{k,p,0}} \right\rbrack}$

This variant can be further simplified by using only one position of thephase rotators and only one position of the beam attenuator, so that the36 pairs of elementary images acquired in 7.12.2.1. can be reduced to asingle one, with a significant reduction in image quality. In this“extreme simplification” case, the indices c,d,r₁,r₂ now take on asingle value and the entire procedure described in 7.12.2. is reducedto:${M_{k,p,q}\left\lbrack {i,j} \right\rbrack} = \frac{{{{{M0}\left\lbrack {k,p} \right\rbrack}\left\lbrack {0,0,0,0} \right\rbrack}\left\lbrack {q,i,j} \right\rbrack} - {{Iref}\left\lbrack {{{\overset{\_}{p}q} + {p\overset{\_}{q}}},i,j} \right\rbrack}}{\sqrt{{Iref}\left\lbrack {{{\overset{\_}{p}q} + {p\overset{\_}{q}}},i,j} \right\rbrack}}$

7.18.7. Simplified method for obtaining three-dimensional images

To generate the three-dimensional image of the object, it possible tolimit the procedure to the representation F_(0,0) defined in 7.17. Inthe procedure described in 7.17.2., this is equivalent to adopting zeroarrays IB_(p,q) for every pair (p,q)≠(0,0). The present method is themethod defined in 7.18.4. but simplified in this manner, and adapted tothe case in which the average index of the sample is close to thenominal index of the objectives, and in which the table can beconsidered perfectly stable.

Under these conditions, we have R_(k,0,0)=1 and D₀[i,j]=1.

The formula:${M_{k,p,q}\left\lbrack {i,j} \right\rbrack} = \frac{{M_{k,p,q}\left\lbrack {i,j} \right\rbrack}{D_{{p\overset{\_}{q}} + {\overset{\_}{p}q}}\left\lbrack {i,j} \right\rbrack}R_{k,p,q}}{{D_{p}\left\lbrack {{imax}_{k,p,0},{jmax}_{k,p,0}} \right\rbrack}R_{k,p,0}}$

which was used in 7.18.4. is thus simplified and we obtain:

M_(K,0,0)[i,j]=M_(k,0,0)[i,j] i.e the array M_(k,0,0) is not modifiedbefore being used to generate the three-dimensional representation ofthe object.

In this particular case, no algorithmic compensation for the phasedifference of the reference beam is required, because the device usedallows the generation of illuminating beams having a constant phasedifference in relation to the reference wave.

In this particular case, it is also possible to obviate Steps 7.10,7.11, 7.15 and 7.16. The position of the objectives can be adjusted asin 7.9.1, so as to have a centered point image. The control of the phaseshift device is then defined by:

Θ_(d)=θ_(d)−Arg(Ra[p,Ic[k,p],Jc[k,p]])

i.e. it does not depend on measurements previously carried out on theobject itself. If a microscope objective having a nominal index equal tothat of a vacuum is used, prior measurements may be carried out in theabsence of any object (considering the transparent plate to be anobject).

7.18.8. Producing an image with a single position of the polarizers

The present variant consists in not using the possibility of varying theindices r₁ and r₂. If images according to the present variant aregenerated alone, the polarization rotators (2210) (2241) (2238) (2226)may be eliminated. The present variant requires a modification of Step 3of the loop on k,p described in 7.12.2.2., as well as a modification ofStep 8 of the algorithm described in 7.17.2.

7.18.8.1. Modification of Step 3 of the loop on k,p described in7.12.2.2.

When the polarization rotators are not used, the direction of theelectric field vector of the illuminating beam and the analysisdirection of the received beam are oriented along the vector {rightarrow over (j)} of FIG. 52.

We denote as {right arrow over (y)}_(e) and {right arrow over (y)}_(c)the vectors deduced respectively from the vectors {right arrow over(x)}_(e) and {right arrow over (x)}_(c) by a rotation of π/2 in theplane of FIG. 52.

We have:

{right arrow over (j)}={right arrow over (x)} _(e) sin φ_(e) +{rightarrow over (y)} _(e) cos φ_(e)

During the diffraction towards point C:

the component along {right arrow over (x)}_(e) is transmitted withoutattenuation, becoming the component on the vector −{right arrow over(x)}_(c)

the component along {right arrow over (y)}_(e) is transmitted, becomingthe component on the vector −{right arrow over (y)}_(c), but attenuatedby a factor cos θ in which θ is the angle between the vector f_(e) andthe vector f_(c).

For an electric field vector {right arrow over (j)} of the illuminatingwave, the electric field vector of the wave received at point C is thusproportional to {right arrow over (t)}=−{right arrow over (x)}_(c) sinφ_(e)−{right arrow over (y)}_(c) cos φ_(e) cos θ. The relationships{right arrow over (x)}_(c)={right arrow over (i)} cos φ_(c)+{right arrowover (j)} sin φ_(c), {right arrow over (y)}_(c)=−{right arrow over (i)}sin φ_(c)+{right arrow over (j)} cos φ_(c) can be taken into account.The component of {right arrow over (t)} along {right arrow over (j)} isalso proportional to Br=−cos φ_(c) cos φ_(e) cos θ−sin φ_(c) sin φ_(e).This constitutes an attenuation factor which affects the diffracted beammeasured at the point C along an analysis direction oriented along{right arrow over (j)} when the illuminating beam is directed on thepoint E and has its electric field vector oriented along {right arrowover (j)}. If the diffusion could be considered isotropic, thiscoefficient would be constant. To compensate for the anisotropic effect,it is thus sufficient to divide the values measured by this coefficientBr so as to arrive at a constant coefficient characterizing theisotropic diffusion. Division by Br raises the level of the noiseaffecting the points at which Br is low, which must also be taken intoaccount.

The factor cos θ is equal to${\cos \quad \theta} = {\frac{f_{e} \cdot f_{c}}{{f_{e}} \cdot {f_{c}}}.}$

Using the normalized values of the frequency vectors, this results in:cos θ=x_(c)x_(e)+y_(c)y_(e)+z_(c)z_(e).

Step 3 of the procedure 7.12.12. is thus modified as follows:

the program calculates, in addition to the values already calculated in7.12., cos θ=x_(c)x_(e)+y_(c)y_(e)+z_(c)z_(e)

the program calculates the attenuation compensation factor comp[k,p,q]:

if −cos φ_(c) cos φ_(e) cos θ−sin φ_(e)≦lim then comp[k,p,q]=1/lim,where lim is a very low value, for example lim=10⁻¹⁰

otherwise,${{comp}\left\lbrack {k,p,q} \right\rbrack} = \frac{1}{{{- \cos}\quad \phi_{c}\quad \cos \quad \phi_{e}\quad \cos \quad \theta} - {\sin \quad \phi_{c}\quad \sin \quad \phi_{e}}}$

Like the values coef[k,p,q,i,j][r₁,r₂] used in 7.12., the valuescomp[k,p,q] constitute an array which can be precalculated.

the program finally calculates:

 M _(k,p,q) [i,j]=M 2[1,1][q,i,j]comp[k,p,q]${b_{k,p,q}\left\lbrack {i,j} \right\rbrack} = {\frac{\sigma_{acq}}{\sqrt{6}}\quad \frac{{comp}\left\lbrack {k,p,q} \right\rbrack}{{{att}\left\lbrack {{M1}\left\lbrack {q,i,j} \right\rbrack} \right\rbrack}\quad \sqrt{{Iref}\left\lbrack {{{\overset{\_}{p}q} + {p\overset{\_}{q}}},i,j} \right\rbrack}}}$

where σ_(acq) is the standard deviation of the noise of the sensors. Thesensors are generally designed so that σ_(acq) is of the order of 1. Wehave: $\sigma_{acq} = 10^{{- \frac{{SNR}_{db}}{20}}\quad 2^{N}}$

 where SNR_(dB) is the signal-to-noise ratio in decibels and N is thenumber of sampling bits of the signal.

7.18.8.2. Modification of Step 8 of the algorithm described in 7.17.2.

Multiplication by comp[k,p,q] can make the noise level riseconsiderably, which can distort the image obtained. To prevent thisproblem, Step 8 of the procedure described in 7.17 is modified so as tocancel the components of the frequency representation which are lower inmodulus than the noise multiplied by a given constant const.

Cancellation of certain elements of the frequency representation isitself a noise generator. To obtain a frequency representation with aquality comparable to that obtained with the normal procedure, moreprecise sampling or an addition attenuation level may be necessary.

Step 8 of the procedure described in 7.17 is modified as follows:

Step 8 modified: The program calculates the final frequencyrepresentation, contained in an array F of dimensions2N_(pix)×2N_(pix)×2N_(pix). It initializes to 0 this array and then goesthrough the indices ni,nj,nk while testing the condition:${\sum\limits_{p,q}{{IB}_{p,q}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}} \neq 0$

 When the condition is met, it performs:${F\left\lbrack {{ni},{nj},{nk}} \right\rbrack} = \frac{\sum\limits_{p,q}{{F_{p,q}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}\quad {{IB}_{p,q}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}}}{\sum\limits_{p,q}\quad {{IB}_{p,q}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}}$

 It then tests the condition${F\left\lbrack {{ni},{nj},{nk}} \right\rbrack} \leq \frac{const}{\sqrt{\sum\limits_{p,q}\quad {{IB}_{p,q}\left\lbrack {{ni},{nj},{nk}} \right\rbrack}}}$

 When the condition is met, it performs: F[ni,nj,nk]=0

const is a constant chosen so that in the absence of a signal (noiseonly) the condition is always true. It is possible, for example, to useconst=4

Multiplication by a constant of the total noise level modifies theresults of this Step 8 modified. For this reason, in 7.18.8.1., anabsolute noise level is determined, whereas in 7.12 the noise level wasdefined with a constant bias.

7.18.9. Obtaining an image of uniaxial birefringent crystal

We consider a uniaxial crystal of ordinary index n_(o) cut so as to forma plate of small thickness, the plane of the plate being orthogonal tothe optical axis of the crystal. This plate constitutes the observedsample and is placed between the two objectives, optical oil being usedbetween the objectives and the sample, the objective being designed forthe use of cover glasses having an index equal to that of the opticaloil. The optical axis of the crystal thus coincides with the opticalaxis of the objectives.

It is assumed that this plate is not “perfect”. It can be affected, forexample, by localized crystallization faults. The purpose of the presentprocedure is to obtain a three-dimensional image of thesecrystallization faults. It may also be an optical memory whose localindex variations characterize the stored bits. The present proceduremakes it possible to obtain a three-dimensional image characteristic ofthe variations of the ordinary index n_(o) of the sample.

The average ordinary index n_(o) is assumed to be known, as is thethickness of the plate. The sample is introduced without displacing theobjectives, so that x,y,z are also known. It is also possible to obtainn_(o),L, x,y,z with a modified version of the procedure described in7.11. Such a modified version, applicable in the case of embodiment 4,will be described in 8.4.3.2.

Obtaining the image of the ordinary index supposes a modification ofStep 8 of the procedure 7.17., which is the one already described in7.18.8.2. It also assumes a modification of Step 3 of the loop on k,pdescribed in 7.12.2.2., described below using the notations used in7.12.

7.18.9.1. Principle

We denote:${{\overset{\rightharpoonup}{u}}_{e} = \frac{f_{e}\bigwedge f_{o}}{{f_{e}\bigwedge f_{o}}}},{{\overset{\rightharpoonup}{u}}_{c} = \frac{f_{c}\bigwedge f_{o}}{{f_{c}\bigwedge f_{o}}}}$

{right arrow over (u)}_(e) and {right arrow over (u)}_(c) are thusdefined according to the vectors of FIG. 51. They are both in the planeof FIG. 52 (not shown). {right arrow over (u)}_(e) is oriented along theordinary polarization direction of the illuminating wave reaching E.{right arrow over (u)}_(c) is oriented along the ordinary polarizationdirection of the diffracted wave reaching the point C.

We denote {right arrow over (v)}_(e) and {right arrow over (v)}_(c) thevectors deduced respectively from the vectors {right arrow over (u)}_(e)and {right arrow over (u)}_(c) by π/2 rotation in the plane of FIG. 52.

We denote {right arrow over (y)}_(e) and {right arrow over (y)}_(c) thevectors deduced respectively from the vectors {right arrow over (x)}_(e)and {right arrow over (x)}_(c) by π/2 rotation in the plane of FIG. 52.

We denote ε the angle between the vector {right arrow over (i)} and thevector {right arrow over (a)} obtained from {right arrow over (j)} by asymmetry whose axis is the position of the neutral axis of thepolarization rotators when this neutral axis is not parallel to {rightarrow over (j)}. We typically have ε≈10 degrees.

When the electric field vector of the illuminating beam (at point E) isA₀{right arrow over (i)}+A₁{right arrow over (j)}, the electric fieldvector measured at the point C is (C₀₀A₀+C₀₁A₁){right arrow over(i)}+(C₁₀A₀+C₁₁A₁){right arrow over (j)}.

When an electric field vector {right arrow over (u)}_(e)={right arrowover (i)} cos α_(e)+{right arrow over (j)} sin α_(e) is used for theilluminating wave, the electric field vector measured at the point C isthus: {right arrow over (w)}_(m)=(C₀₀ cos α_(e)+C₀₁ sin α_(e)){rightarrow over (i)}+(C₁₀ cos α_(e)+C₁₁ sin α_(e)){right arrow over (j)}

We can use:

{right arrow over (i)}={right arrow over (u)} _(c) cos α_(c) −{rightarrow over (v)} _(c) sin α_(c)

{right arrow over (j)}={right arrow over (u)} _(c) sin α_(c) −{rightarrow over (v)} _(c) cos α_(c)

giving:

{right arrow over (w)} _(m)=(C ₀₀ cos α_(e) cos α_(c) +C ₀₁ sin α_(e)cos α_(c) +C ₁₀ cos α_(e) sin α_(c) +C ₁₁ sin α_(e) sin α_(c)){rightarrow over (u)} _(c)

+(−C ₀₀ cos α_(e) sin α_(c) −C ₀₁ sin α_(e) sin α_(c) +C ₁₀ cos α_(e)

cos α_(c) +C ₁₁ sin α_(e) cos α_(c)){right arrow over (v)} _(c)

The value measured at the point C along the direction of the vector{right arrow over (u)}_(c) is thus:

Bm=C ₀₀ cos α_(e) cos α_(c) +C ₀₁ sin α_(e) cos α_(c) +C ₁₀ cos α_(e)sin α_(c) +C ₁₁ sin α_(e) sin α_(c)

We denote Q_(r) ₁ _(,r) ₂ the value measured at the point C for thecombination r₁,r₂ of the control indices of the polarization rotators.When the electric field vector of the illuminating beam (at the point E)is A₀{right arrow over (a)}+A₁{right arrow over (j)}, the electric fieldvector measured at the point C is (Q₀₀A₀+Q₀₁A₁){right arrow over(a)}+(Q₁₀A₀+Q₁₁A₁){right arrow over (j)} with {right arrow over(a)}={right arrow over (i)} cos ε+{right arrow over (j)} sin ε.

The relationship defining {right arrow over (a)} is inverted as:$\overset{\rightharpoonup}{i} = {{\overset{\rightharpoonup}{a}\quad \frac{1}{\cos \quad ɛ}} + {\overset{\rightharpoonup}{j}\quad \tan \quad ɛ}}$

When the electric field vector of the illuminating beam is${{A_{0}\quad \overset{\rightharpoonup}{i}} + {A_{0}\quad \overset{\rightharpoonup}{i}}} = {{\overset{\rightharpoonup}{a}\quad A_{0}\quad \frac{1}{\cos \quad ɛ}} + {\overset{\rightharpoonup}{j}\quad \left( {{{- A_{0}}\quad \tan \quad ɛ} + A_{1}} \right)}}$

the vector received at the point C is thus:${\left( {{Q_{00}\quad A_{0}\quad \frac{1}{\cos \quad ɛ}} + {Q_{01}\quad \left( {{{- A_{0}}\quad \tan \quad ɛ} + A_{1}} \right)}} \right)\quad \overset{\rightharpoonup}{a}} + {\left( {{Q_{10}\quad A_{0}\quad \frac{1}{\cos \quad ɛ}} + {Q_{11}\quad \left( {{{- A_{0}}\quad \tan \quad ɛ} + A_{1}} \right)}} \right)\quad \overset{\rightharpoonup}{j}}$

hence:${\left( {{Q_{00}\quad A_{0}} + {Q_{01}\quad \left( {{{- A_{0}}\quad \sin \quad ɛ} + {A_{1}\quad \cos \quad ɛ}} \right)}} \right)\quad \overset{\rightharpoonup}{i}} + {\left( {{Q_{00}\quad A_{0}\quad \tan \quad ɛ} + {Q_{01}\quad \left( {{{- A_{0}}\quad \sin \quad ɛ\quad \tan \quad ɛ} + {A_{1}\quad \sin \quad ɛ}} \right)} + {Q_{10}\quad A_{0}\quad \frac{1}{\cos \quad ɛ}} + {Q_{11}\quad \left( {{{- A_{0}}\quad \tan \quad ɛ} + A_{1}} \right)}} \right)\quad \overset{\rightharpoonup}{j}}$

This expression is the equivalent of (C₀₀A₀+C₀₁A₁){right arrow over(i)}+(C₁₀A₀+C₁₁A₁){right arrow over (j)}, with:

C ₀₀ =Q ₀₀ −Q ₀₁ sin ε

C ₀₁ =Q ₀₁ cos ε

$C_{10} = {{Q_{00}\quad \tan \quad ɛ} - {Q_{01}\quad \sin \quad ɛ\quad \tan \quad ɛ} + {Q_{10}\quad \frac{1}{\cos \quad ɛ}} - {Q_{11}\quad \tan \quad ɛ}}$

 C ₁₁ =Q ₀₁ sin ε+Q ₁₁

The expression of Bm is thus transformed as follows:${Bm} = {{\left( {Q_{00} - {Q_{01}\quad \sin \quad ɛ}} \right)\quad \cos \quad \alpha_{e}\quad \cos \quad \alpha_{c}} + {Q_{01}\quad \cos \quad ɛ\quad \sin \quad \alpha_{e}\quad \cos \quad \alpha_{c}} + {\left( {{Q_{01}\quad \sin \quad ɛ} + Q_{11}} \right)\quad \sin \quad \alpha_{e}\quad \sin \quad \alpha_{c}} + {\left( {{Q_{00}\quad \tan \quad ɛ} - {Q_{01}\quad \sin \quad ɛ\quad \tan \quad ɛ} + {Q_{10}\quad \frac{1}{\cos \quad ɛ}} - {Q_{11}\quad \tan \quad ɛ}} \right)\quad \cos \quad \alpha_{e}\quad \sin \quad \alpha_{c}}}$

hence:${Bm} = {{\left( {{\cos \quad \alpha_{e}\quad \cos \quad \alpha_{c}} + {\cos \quad \alpha_{e}\quad \sin \quad \alpha_{c}\quad \tan \quad ɛ}} \right)\quad Q_{00}} + {\left( {{{- \sin}\quad ɛ\quad \cos \quad \alpha_{e}\quad \cos \quad \alpha_{c}} + {\cos \quad ɛ\quad \sin \quad \alpha_{e}\quad \cos \quad \alpha_{c}} + {\sin \quad ɛ\quad \sin \quad \alpha_{c}} - {\sin \quad ɛ\quad \tan \quad ɛ\quad \cos \quad \alpha_{e}\quad \sin \quad \alpha_{c}}} \right)\quad Q_{01}} + {\left( {\frac{1}{\cos \quad ɛ}\quad \cos \quad \alpha_{e}\quad \sin \quad \alpha_{c}} \right)\quad Q_{10}} + {\left( {{\sin \quad \alpha_{e}\quad \sin \quad \alpha_{c}} - {\tan \quad ɛ\quad \cos \quad \alpha_{e}\quad \sin \quad \alpha_{c}}} \right)\quad Q_{11}}}$

This value Bm is the value measured at the point C along the directionof the vector {right arrow over (u)}_(c) when the electric field vectorof the illuminating wave is oriented along {right arrow over (u)}_(e).Owing to the definition of the vectors {right arrow over (u)}_(e) and{right arrow over (u)}_(c), Bm is the value of the ordinary diffractedray for an ordinary illuminating beam. The measurement of Bm is indirectsince it is the values Q_(ij) which are measured, the value of Bm beingdeduced therefrom.

The vector {right arrow over (u)}_(e) may be expressed in the followingform:

{right arrow over (u)} _(e) ={right arrow over (x)} _(e) cos β_(e)−{right arrow over (y)} _(e) sin β_(e)

During diffraction towards the point C, on the same principle as in7.18.8.1.:

the component along {right arrow over (x)}_(e) is transmitted withoutattenuation, becoming the component on the vector −{right arrow over(x)}_(c)

the component along {right arrow over (y)}_(e) is transmitted, becomingthe component on the vector −{right arrow over (y)}_(c), but isattenuated by a factor cos θ where θ is the angle between the vectorf_(e) and the vector f_(c).

The vector received at the point C is thus:

{right arrow over (w)} _(r) =−{right arrow over (x)} _(c) cos β_(e)+{right arrow over (y)} _(c) sin β_(e) cos θ

We can use:

{right arrow over (x)} _(c) ={right arrow over (u)} _(c) cos β_(c)+{right arrow over (v)} _(c) sin β_(c)

{right arrow over (y)} _(c) =−{right arrow over (u)} _(c) sin β_(c)+{right arrow over (v)} _(c) cos β_(c)

We consequently obtain:

{right arrow over (w)} _(r)=(−cos β_(c) cos β_(e)−sin β_(c) sin β_(e)cos θ){right arrow over (u)} _(c)+(−sin β_(c) cos β_(e)+cos β_(c) sinβ_(e) cos θ){right arrow over (v)} _(c)

The value received at the point C along the direction of the vector{right arrow over (u)}_(c) is thus:

Br=−cos β_(c) cos β_(e)−sin β_(c) sin β_(e) cos θ

This value Br is the attenuation coefficient which affects the ordinarydiffracted beam received at the point C when the illuminating beam isordinary. In an isotropic diffusion model, the coefficient Br would beconstant. As in 7.18.8.1., this attenuation may be compensated bydividing the measured value Bm by the coefficient Br.

7.18.9.2. Algorithm

Step 3 of the loop on k,p described in 7.12.12. is thus modified asfollows:

the program calculates the following quantities, in addition to thosecalculated in 7.12.12.:

cos θ=x _(c) x _(e) +y _(c) y _(e) +z _(c) z _(e)${\sin \quad \beta_{c}} = {{\frac{1}{M_{c}\quad M_{ce}}\quad V_{{xy}\quad}\quad \cos \quad \beta_{c}} = {\frac{1}{M_{c}\quad M_{ce}}\quad \left( {{y_{c}\quad V_{yz}} - {x_{c}\quad V_{xz}}} \right)}}$${\sin \quad \alpha_{c}} = {{\frac{1}{M_{c}}\quad x_{c}\quad \cos \quad \alpha_{c}} = {\frac{1}{M_{c}}\quad y_{c}}}$${\sin \quad \beta_{e}} = {{\frac{1}{M_{e}\quad M_{ce}}\quad V_{{xy}\quad}\quad \cos \quad \beta_{e}} = {\frac{1}{M_{e}\quad M_{ce}}\quad \left( {{{- y_{e}}\quad V_{yz}} + {x_{e}\quad V_{xz}}} \right)}}$${\sin \quad \alpha_{e}} = {{\frac{1}{M_{e}}\quad x_{e}\quad \cos \quad \alpha_{e}} = {\frac{1}{M_{e}}\quad y_{e}}}$

 For all these values, appropriate limit values are used when thedenominators are zero:

if M_(c)=0 then α_(c)=−α_(e) and β_(c)=β_(e)=0 is used

if M_(e)=0 then α_(e)=−α_(c) and β_(c)=β_(e)=0 is used

if M_(ce)=0 then β_(c)=β_(e)=0 is used

if M_(c)=0 and M_(e)=0 and M_(ce)=0 then α_(c)=α_(e)=β_(c)=β_(e)=0 isused

the program calculates the attenuation compensation factor comp[k,p,q]:

if −cos β_(c) cos β_(e)−sin β_(c) sin β_(e) cos θ≦lim, then${{{comp}\left\lbrack {k,p,q} \right\rbrack} = \frac{1}{\lim}},$

 where lim is a very low value, for example lim=10 ⁻¹⁰

otherwise,${{comp}\left\lbrack {k,p,q} \right\rbrack} = \frac{1}{{{- \cos}\quad \beta_{c}\quad \cos \quad \beta_{e}}\quad - {\sin \quad \beta_{c}\quad \sin \quad \beta_{e}\quad \cos \quad \theta}}$

The program calculates the values:

coef[k,p,q,i,j][0,0]=(cos α_(e) cos α_(c)+cos α_(e) sin α_(c) tanε)comp[k,p,q]

coef[k,p,q,i,j][0,1]=(−sin ε cos α_(e) cos α_(c)+cos ε sin α_(e) cosα_(c)+sin ε sin α_(e) sin α_(c)−sin ε tan ε cos α_(e) sinα_(c))comp[k,p,q]${{{coef}\left\lbrack {k,p,q,i,j} \right\rbrack}\left\lbrack {1,0} \right\rbrack} = {\left( {\frac{1}{\cos \quad ɛ}\quad \cos \quad \alpha_{e}\quad \sin \quad \alpha_{c}} \right)\quad {{comp}\left\lbrack {k,p,q} \right\rbrack}}$

 coef[k,p,q,i,j][ 1,1]=(sin α_(e) sin α_(c)−tan ε cos α_(e) sinα_(c))comp[k,p,q]

As in 7.12., the values coef[k,p,q,i,j][r₁,r₂] and comp[k,p,q]constitute arrays which can be precalculated.

the program finally calculates:${M_{k,p,q}\left\lbrack {i,j} \right\rbrack} = {\sum\limits_{r_{1},r_{2}}{{{{M2}\left\lbrack {r_{1},r_{2}} \right\rbrack}\left\lbrack {q,i,j} \right\rbrack}\quad {{{coef}\left\lbrack {k,p,q,i,j} \right\rbrack}\left\lbrack {r_{1},r_{2}} \right\rbrack}}}$${B_{k,p,q}\left\lbrack {i,j} \right\rbrack} = {\frac{\sigma_{acq}}{\sqrt{6}}\quad \frac{{comp}\left\lbrack {k,p,q} \right\rbrack}{{{att}\left\lbrack {{M1}\left\lbrack {q,i,j} \right\rbrack} \right\rbrack}\sqrt{{Iref}\left\lbrack {{{\overset{\_}{p}q} + {p\overset{\_}{q}}},i,j} \right\rbrack}}}$

where σ_(acq) is the standard deviation of the sensor noise defined asin 7.18.8.1. M2[r₁,r₂][q,i,j] corresponds to the value measured whichwas denoted Q_(r) ₁ _(,r) ₂ in 7.18.9.1. the calculation ofM_(k,p,q)[i,j] is equivalent to the calculation of the ratio$\frac{Bm}{Br}$

 defined in 7.18.9.1.

7.18.10. Study of samples by reflection only

When the two representations F_(0,0) and F_(0,1) do not match, it is inprinciple impossible to determine their phase relationship: in fact, anyphase relationship corresponds to a possible frequency representation. Apossible solution is then to obtain the representation of the objectfrom F_(0,1) alone. However, with information on F_(0,1) alone it is notpossible to reliably differentiate the refractive index andabsorptivity. In fact, a representation differentiating validly therefractive index and absorptivity must occupy a region of the frequencyspace symmetrical in relation to the origin. A good qualityrepresentation, comparable to the one obtained with confocal microscopyby reflection but with a higher precision can be obtained by extractingthe modulus from the spatial representation obtained by using F_(0,1)alone.

This method can also be applied to the study of the surface area ofthick samples through which the beam is transmitted poorly. In thiscase, the approximation whereby the illumination is roughly constant inthe observed region of the sample is not complied with in the surfacearea adjacent to the microscope objectives, and when the illuminatingbeam and the wave coming from the object are transmitted by the sameobjective. The only reliable part of the frequency representationobtained is thus F_(0,1) if the top side of the object is observed, orF_(1,1) if the bottom side of the object is observed. As previously, itis possible to extract the modulus from the spatial representationobtained using only F_(0,1) or F_(1,1).

In a similar manner, this method can be used for the study of thesurface of very opaque objects.

It is also possible to construct, on this principle, a degraded versionof the microscope having only one objective and hence operating only bereflection. The adjustment phase must then be adapted so that a secondobjective is not required. It is also possible to use a complete systemduring the adjustment phase and then eliminate the second objective onlylater.

7.19. Use of microscope objectives with a suitable nominal index

In all the embodiments, it is possible to use standard microscopeobjectives. These objectives have a nominal refractive index n_(v) closeto that of glass. They are designed to operate with an immersion liquidand a cover glass of index n_(v). These objectives yield good results ifthe observed sample has an average index close to that of n_(v) or isnot very thick. If the observed sample consists essentially of water, ofindex 1.33, and if the aperture of the objective is 1.4, then the totalthickness of the sample must be sufficiently smaller than the totalwidth of the generated image. If it is too high, the three-dimensionalrepresentation obtained may be distorted. In fact, the sphericalaberration caused by the thickness of the sample may then become suchthat the wave coming from an objective cannot be received by the facingobjective except for a small part of the frequencies used.

A microscope objective is designed to use an optical liquid with a givenindex, the index of the optical liquid having been called here the<<nominal index>> of the objective. It is also designed to use a coverglass having a given index and thickness, the thickness of the coverglass not being necessarily equal to the nominal index of the objective.If the average index of the object is assumed to be equal to the nominalindex of the objective, the objective allows a compensation for thespherical aberration due to the cover glass, which is independent of theposition of the object.

If, on the other hand, the nominal index of the objective differs fromthe average index of the object, the variations in the position of theobject lead to a variation in the thickness of the layers correspondingrespectively to the object and to the optical liquid. The inducedspherical aberration thus depends on the position of the object and canconsequently not be compensated by the objective, a compensation validfor a position of the object no longer being so for another position.This problem has no effect on the possibility of obtainingtwo-dimensional images, which is usual with conventional microscopes,and in which the index of the sample does not intervene. By contrast, itis troublesome for the observation of images in three dimensions with athick sample.

A solution to this problem consists in using an objective whose nominalindex is close to the average index of the observed sample.

If the observed sample consists essentially of water, the optical liquidmay be water or a liquid having a stabilized index close to that ofwater. The objective can be designed by usual optical calculationmethods, taking into account the index of the optical liquid and theneed to compensate for aberration due to the cover glass. This designmay be facilitated by the use of a Teflon cover (polymer manufactured byDuPont) whose index is close to that of water and which thus induces asmall aberration.

If the observed sample is a birefringent crystal with a high index, asin the case of optical memories, the nominal index of the objectiveshould be close to that of the crystal, implying the use of a high indexoptical liquid. The cover glass must have an index close to that of theobserved crystal or may be eliminated.

Water-immersion objectives are, for example, manufactured by Zeiss.

7.20. Use of microscope objectives affected by spherical aberration

FIG. 90 represents the plane (6105) whose image is formed by themicroscope objective in (6107) and the rear focal plane (6106) of theobjective. We denote as B(α) the image point, in the rear focal plane ofthe objective, of a beam forming in the object an angle α with theoptical axis.

The present variant consists in using objectives designed so as toverify only the following constraints:

(1)—Spherical aberration affecting the image formed in the image planeof the objective must remain less than a fraction of the diameter ofthis image.

(2)—The image, in the rear focal plane, of a beam which is parallel inthe observed object, must be a point image.

(3)—The distance between the point B(α) and the point B(0) must beproportional to sin α.

An objective complying only with these constraints is easier to designthan a conventional objective. This simplification of requirementsimposed on the objective makes it possible to increase the workingdistance, thus allowing the object to be placed between thick plates. Itcan also allow an increase in the aperture of the objectives, thusfacilitating compliance with the matching condition for partialfrequency representations. An objective of this type may consist, forexample, of a Nikon aplanetic/achromatic condenser with an aperture of1.4 associated with an achromat allowing its magnification to beadjusted. FIG. 89 shows the principle of such an association. The Nikoncondenser is designed for a parallel incoming light. The beam comingfrom a point (6104) of the observed object is thus parallel to theoutput of the condenser (6102). An achromat (6101) is used to make itconverge again towards a point (6103). If the focal length of thecondenser is f_(c) and if the focal length of the achromat is f_(a) thenthe magnification of the combination is f_(a)/f_(c) and the distancebetween the achromat and the image plane, where the diaphragm should belocated, is f_(a).

Conditions (2) and (3) must be interpreted as meaning that thedeviations with regard to punctuality or errors in the position of thepoint in the rear focal plane are of an amplitude smaller than thediameter of the corresponding diffraction disc, which is about$\frac{D}{N_{pix}}$

where:

N_(pix) is the number of sampling pixels of the plane frequency imagesused.

D is the diameter of the image formed in the rear focal plane of theobjective.

An objective complying with requirements (1) to (3) can be designed bymeans of an optical calculation program. In this case, the programdetermines the optical paths. It can in particular determine the lengthof the optical path of the ray between point A and point B(α). Thislength is denoted chem(sin α).

The phase difference, in the rear focal plane, of the corresponding beamis then$2\quad \pi \quad {\frac{{chem}\quad \left( {\sin \quad \alpha} \right)}{\lambda}.}$

This phase shift can be compensated for by multiplying the point ofcoordinates (i,j) of the plane frequency image by:$\exp \quad \left( {{- \overset{\sim}{j}}\quad \frac{2\quad \pi}{\lambda}\quad {chem}\quad \left( {\frac{1}{K}\quad \sqrt{\left( {i - \frac{N_{pix}}{2}} \right)^{2} + \left( {j - \frac{N_{pix}}{2}} \right)^{2}}} \right)} \right)$

This can be done by taking into account this phase difference in theaberration compensation function obtained in 7.16. If the value chem(sinα) cannot be determined by optical calculation, it can also bedetermined by a measurement process.

In the following explanations, the value chem(sin α) will always be usedin functional form. It is however clear that it can in practice be inthe form of arrays. Oversampling/undersampling methods can be used toobtain it in the form of arrays of different dimensions or havingdifferent sampling intervals. The objectives (2217) and (2219) must inevery case be identical to each other.

The use of objectives complying only with conditions (1) (2) (3) callsfor certain modifications in the method used.

7.20.1. Modification of method for calculating coefficients K_(p)described in 7.6.

Since there is significant spherical aberration, it is not possible tofocus the image on the micrometer as indicated in 7.6. The focussingprocedure and the calculation formula for K_(p) from the image of themicrometer are consequently eliminated and the coefficients K_(p) arecalculated directly from the frequency image.

The object is a micrometer illuminated by a plane wave and characterizedby the distance D_(reel) between two successive graduations. The wavecoming from the object has a maximum intensity for angles formed withthe optical axis complying with${\sin \quad \alpha} = \frac{n\quad \lambda_{v}}{D_{reel}}$

in which n is an integer. The corresponding spatial frequenciesconsequently have as component along the horizontal axis:$\frac{n}{D_{reel}}.$

Moreover, the frequency step

on the image S[p,i,j] is $\frac{1}{K_{p}\quad \lambda_{v}}.$

To measure the coefficient K₀ we use, as object, the objectivemicrometer and the beams FEG and FRD. From the sensor (2239), an imageS[0,i,j] is obtained by the procedure described in 7.5. This image iscomposed of a central point made up of the direct illuminating wave anda series of aligned points corresponding to the different frequenciesfor which the wave coming from the object is at its maximum. Thevisibility of these aligned points is better if the sensor is allowed tosaturate for the central frequency. In this image, we measure thedistance in pixels D_(fr) between two points separated by N_(fr)intervals.

We have, given the preceding considerations:$\frac{D_{fr}}{K_{0}\quad \lambda_{v}} = \frac{N_{fr}}{D_{reel}}$

or, with$K_{0} = {\frac{D_{reel}\quad D_{fr}}{\lambda_{v}\quad N_{fr}}.}$

The wavelength to be considered here is the wavelength in the material,assumed to have an index equal to the nominal index n_(v) of theobjective, i.e.: $\lambda_{v} = {\frac{\lambda}{n_{v}}.}$

We thus have finally:$K_{0} = {\frac{n_{v}}{\lambda}\quad \frac{D_{reel}\quad D_{fr}}{N_{fr}}}$

K₁ is measured by a symmetrical process from sensor (2229).

7.20.2. Modification of procedure described in 7.9.1. for obtaining therelative coordinates of objectives

When the procedure described in 7.9.1. is applied, the image obtained onthe CCD (2239) is affected by a phase shift which is twice that whichwould affect the wave coming from a point of the sample. In fact, thebeam FRGI passes through two objectives instead of one. This phaseshift, which would not exist if the objectives were devoid of sphericalaberration, must be compensated to obtain a function F_(rec) from whichit is possible to apply the program described in 7.8.

Before determining the coordinates x,y,z by means of the programdescribed in 7.8., the array F_(rec) obtained must consequently bemodified as follows:${F_{rec}\left\lbrack {i,j} \right\rbrack} = {{F_{rec}\left\lbrack {i,j} \right\rbrack}\quad \exp \quad \left( {{- 2}\quad \frac{2\quad \pi}{\lambda}\quad {chem}\quad \left( {\frac{1}{K_{0}}\quad \sqrt{\left( {i - \frac{N_{pix}}{2}} \right)^{2} + \left( {j - \frac{N_{pix}}{2}} \right)^{2}}} \right)} \right)}$

7.20.3. Modification of procedure described in 7.15.2. for calculatingw_(p)

In order to be able to calculate w_(p), it is essential to compensate,for each plane frequency image, the phase shifts due to the sphericalaberrations of the microscope objectives.

In the block (4002) of FIG. 57, an additional step 1.4. must be addedafter Step 1.3. Step 1.4.: the program goes through the indices i,jperforming:${{Ms}_{k}\left\lbrack {i,j} \right\rbrack} = {{{Ms}_{k}\left\lbrack {i,j} \right\rbrack}\quad \exp \quad \left( {{- \frac{2\quad \pi}{\lambda}}\quad {chem}\quad \left( {\frac{N_{pix}}{K_{0}\quad N_{d}}\quad \sqrt{\left( {i - \frac{N_{d}}{2}} \right)^{2} + \left( {j - \frac{N_{d}}{2}} \right)^{2}}} \right)} \right)}$

7.20.4. Modification of aberration compensation function calculationdescribed in 7.16.

The compensation for spherical aberration of the objectives is obtainedby a corresponding modification of the aberration compensation functionD_(p).

After Step 1 of the procedure described in 7.16., it is necessary to addan additional step: Step 1 bis: the program goes through all the indicesi,j performing the operation:${{Ds}_{p}\left\lbrack {i,j} \right\rbrack} = {{{Ds}_{p}\left\lbrack {i,j} \right\rbrack}\quad \exp \quad \left( {{- \frac{2\quad \pi}{\lambda}}\quad {chem}\quad \left( {\frac{N_{pix}}{K_{p}\quad N_{e}}\quad \sqrt{\left( {i - \frac{N_{e}}{2}} \right)^{2} + \left( {j - \frac{N_{e}}{2}} \right)^{2}}} \right)} \right)}$

7.20.5. Measurement of chem function

If the chem function is not known through the use of an opticalcalculation program, it can be measured by the microscope. Thismeasurement must be carried out immediately after the measurement of thecoefficients K_(p). To carry out this measurement, we use the beams FRGIand FED. We first use the procedure described in 7.3.3.1. to calculatethe image in the spatial domain and evaluate its punctuality and theposition of its center. The position of the objectives is then adjustedto have the best possible point image and so that this image isperfectly centered. One then determines the image received on the sensor(2239) by means of the procedure described in 7.5. As in 7.20.2., theimage is affected by a phase shift which is twice that which wouldaffect the image of a point of the object and we thus have roughly:${S\left\lbrack {0,i,j} \right\rbrack} = {\exp \left\{ {2\quad \frac{2\quad \pi}{\lambda}\quad {chem}\quad \left( {\frac{1}{K_{0}}\quad \sqrt{\left( {i - \frac{N_{pix}}{2}} \right)^{2} + \left( {i - \frac{N_{pix}}{2}} \right)^{2}}} \right)} \right\}}$

From the array S measured it is possible to reconstitute the chemfunction. A simple way to obtain this function, for a set of samplingpoints indexed by the integer p≦K, is through the equation:${{chem}\quad \left( \frac{p}{K_{0}} \right)} = {\frac{1}{2}\quad \frac{\lambda}{2\quad \pi}\quad {\sum\limits_{i = 1}^{p}\quad {{Arg}\left\{ \frac{S\left\lbrack {0,{i + \frac{N_{pix}}{2}},0} \right.}{S\left\lbrack {0,{i - 1 + \frac{N_{pix}}{2}},0} \right\rbrack} \right\}}}}$

for 1≦p≦K and chem(0)=0, where Arg designates the argument and takes onvalues between −π and π.

The values of chem are obtained here only from measurements carried outon a horizontal straight line. It is possible to use more elaboratemethods to reduce, by filtration, the effect of local disturbances,taking into account all the points and not only those located on such aline.

The chem function thus obtained can be presented in the form of an arrayand oversampled or undersampled as indicated earlier.

7.21. Use of objectives exhibiting spherical aberration and frequencydistortion

The present variant consists in using objectives complying only with theproperties (1) and (2) discussed in 7.20., namely:

(1)—Spherical aberration affecting the image formed in the image planeof the objective must remain less than a fraction of the diameter ofthis image.

(2)—The image, in the rear focal plane, of a beam which is parallel inthe observed object must be a point image.

Such an objective may be formed in the same manner as previously, but itcan then be used under broader conditions, for example with a highernumber of pixels N_(pix). It can also be formed more simply than in thepreceding case.

The fact that we have obviated the property (3) facilitates further theconstruction of the objective. By contrast, the compensation foraberrations induced through non-compliance with property (3) leads to anadditional complexity of the algorithms. Non-compliance with theproperty (3) is such that the coefficient of proportionality between thecoordinates in pixels and the horizontal components of the spatialfrequency is not constant. The coordinates i and j in pixels of thepoints obtained on the CCD sensor must be multiplied by a coefficientA(r) depending on the distance in pixels r between the considered pointand the optical center. The method used thus differs from the onedescribed in 7.20. in the following aspects:

a specific procedure is used to determine A(r).

the plane frequency image which was obtained directly from intensitiesreceived at each point of the CCD sensor CCD must be modified to beutilizable, in each step of the procedure in which it is used.

7.21.1. Calculation of A(r).

To determine the coefficient A(r) an objective micrometer is used as theobject. The point of direct impact of the illuminating beam must matchstrictly with the optical center. As shown in FIG. 91, the imageobtained on the CCD sensor in the absence of a reference wave consistsof a central point P₀ coinciding with the optical center and a series ofaligned points of intensity lower than P₀. We denote as P_(n) thosepoints which are on a given half-line originating at P₀, the maximumvalue of the index n being denoted N. We denote as D_(n) the distance inpixels between the point P_(n) and the optical center. We then have:${A\quad \left( \frac{D_{n} + D_{n + 1}}{2} \right)} = {\frac{N}{D_{N}}\quad {\left( {D_{n + 1} - D_{n}} \right).}}$

This equation gives the values of A at a limited number of points and anoversampling method must be used to obtain the value of A at asufficient number of points.

7.21.2. Modification of Plane frequency images obtained

We denote as S_(brut)[i, j] the array representing a plane frequencyimage obtained for example as indicated in 7.5. S_(brut)[i, j] must bemodified to generate an array S_(fin)[i, j] which will really correspondto the plane frequency image. This modification may be carried out asfollows:

Generation of an array S1, of dimensions N_(sur)×N_(sur) with, forexample, N_(sur)=2048. This array is initialized to 0 and then theprogram carries out, for i and j ranging from 0 to N_(pix)−1:${{S1}\left\lbrack {{A\quad \left( {i\quad \frac{N_{sur}}{N_{pix}}} \right)},{A\quad \left( {j\quad \frac{N_{sur}}{N_{pix}}} \right)}} \right\rbrack} = {S_{brut}\left\lbrack {i,j} \right\rbrack}$

Inverse Fourier transformation of array S1 leading to array S2.

Extraction of the middle part of the array S2 to obtain an array S3 withdimensions of N_(pix)×N_(pix). The program carries out, for i and jranging from 0 to N_(pix)−1:${{S3}\left\lbrack {i,j} \right\rbrack} = {{S2}\left\lbrack {{i - \frac{N_{pix}}{2} + \frac{N_{sur}}{2}},{j - \frac{N_{pix}}{2} + \frac{N_{sur}}{2}}} \right\rbrack}$

Fourier transformation of array S3. The array thus obtained is the arrayS_(fin)[i, j] which constitutes the plane frequency image to be used inall the operations.

All the raw plane frequency images obtained in each step of thethree-dimensional adjustment and calculation procedure must be modifiedin this manner before being incorporated in the calculations.

8. FOURTH (PREFERRED) EMBODIMENT

This embodiment is considered to be the best because, in the visibledomain, it is the one that yields the best performance in terms of speedand image quality.

8.1. Principles

The fourth embodiment differs from the third:

in the use of a different beam deflection device.

in the introduction of an additional device allowing the elimination ofthe direct wave reaching the CCDs to prevent or limit the saturationeffect.

in the fact that sampling is “regular”, i.e. the point image of anilluminating beam on the CCD coincides with the center of a pixel of theCCD.

Beam deflection and direct wave suppression devices are based on the useof a spatial modulator (SLM: spatial light modulator) marketed by thecompany Displaytech. This modulator consists of a matrix of 256×256elements each functioning as an independent polarization rotator. Itoperates in reflection, i.e. the light incident on the SLM is reflectedwith a modified polarization, said polarization modification beingdifferent at each point of the matrix. There are two versions: oneintended for amplitude modulation, in which, for one of the controlvoltages, the neutral axis of the ferroelectric liquid crystal (FLC:ferroelectric liquid crystal) is oriented in the direction defined byone of the axes of the matrix, and another intended for phasemodulation, in which the two possible positions of the neutral axis ofthe FLC are symmetrical in relation to one of the axes of the matrix.

Obtaining regular sampling and good control of the beam path alsorequires appropriate use of lenses.

8.1.1. Beam path control

When a plane beam not directed along the optical axis moves away fromits origin, it moves away from the optical axis and can becomenon-utilizable. FIG. 67 illustrates a method making it possible tocontrol the position of such a beam in relation to the optical axis. Aparallel beam (4800) coming from a plane (4801) must be used in a plane(4804) distant from (4801). If it propagates in a straight line, itmoves away from the optical axis and becomes non-utilizable (4805).

The lenses (4802) and (4803) have the same focal length f The plane(4801) is the front focal plane of (4802). (4804) is the rear focalplane of (4803). The front focal plane of (4803) coincides with the rearfocal plane of (4802) and is represented by broken lines (4806).

In the plane (4801), the beam (4800) is parallel and centered on theoptical axis, i.e. its intersection with this plane forms a disccentered on the optical axis. Such a plane will be called <<spatialplane>> and will be denoted by the letter E.

In the plane (4806) the beam is punctual, i.e. its intersection with theplane is practically reduced to a point. Such a plane will be called<<frequency plane>> and will be denoted by the letter F.

In the plane (4804), the beam is again centered and parallel. This planeis thus a new spatial plane. It is the image of the plane (4801) throughthe optical system consisting of lenses (4802) and (4803).

The device makes it possible to reform in the plane (4804) a beamequivalent to the one present in the plane (4801), but symmetrized inrelation to the optical axis.

By modifying the focal length of the second lens as in FIG. 68, it ispossible to modify the angle of the beam in relation to the optical axisand its section. The focal length of the first lens is f₁, that of thesecond lens is f₂, and the distance between the two lenses is f₁+f₂ .The angle of the beam in relation to the optical axis is multiplied byf₁/f₂ and the section of the beam is multiplied by f₂/f₁.

8.1.2. Beam deflection device

A direction at the exit of the deflection device is equivalent to agiven spatial frequency and the terms “frequency” or “angle” will beused hereafter to define a deflection.

The beam deflection device uses a phase SLM whose entire surface isilluminated by a plane beam. When a phase profile (4601) of square-waveform as shown in FIG. 64 is applied to such a surface, the long-distancediffracted intensity is at its maximum for angles α and −α in which α issuch that $h = \frac{\lambda}{2}$

or d sin α=λ/2. These two angles define two symmetrical diffracted beams(4602) and (4603) coming from the SLM. The number of pixels of the SLMused being N_(s) and the step (distance between two pixels) p_(s), sin αvaries from 0 to $\frac{\lambda}{2\quad p_{s}}$

in steps of $\frac{\lambda}{N_{s}\quad p_{s}}$

or a total of $\frac{N_{s}}{2}$

possible values excluding zero. This principle is applied to generate abeam of given direction. This simple device is however not sufficientfor the following reasons:

An attempt is made to generate a single frequency and it is thusnecessary to then eliminate one of the two beams generated, for example(4602)

The beam coming from this simple modulation system is noisy, because, inaddition to the frequency corresponding to the maximum illumination,many spurious frequencies are present.

To eliminate the spurious frequencies and the symmetrical beam, use ismade of a system whose schematic diagram is shown in FIG. 65. In thisdiagram, the SLMs have been shown as if they functioned by transmission,and the polarizers associated with these SLMs have not been represented.A plane beam (4611) incident on a phase SLM (4612) functioning asindicated above is diffracted in two directions represented in solidlines and in broken lines. (4612) is in the front focal plane of a lens(4613). In the rear focal plane of (4613), a beam of a given angle atthe exit of (4612) gives a point image. The front focal plane of thelens (4612) is a spatial plane, and the rear focal plane of (4612) is afrequency plane. In the frequency plane is placed:

a diaphragm (4615) whose functionality is to stop the symmetrical beam(broken line).

an SLM (4614) whose functionality is to suppress the spuriousfrequencies. When the SLM (4612) is controlled to generate a givenfrequency, this frequency corresponds to a point of the SLM (4614). Thispoint is controlled to allow the beam to pass, and the other points ofthe SLM (4614) are controlled to stop the beam. Spurious frequencies arethus eliminated.

A second lens (4616) then transforms again the point obtained in thefrequency plane in a corresponding direction at the exit of the device.

8.1.3. Direct wave elimination device

The wave having traversed the objected and reaching the CCD exhibitsfrequencies of high intensity around the point of direct impact of thebeam. In the preceding embodiment, this led to a saturation of the CCDwhen a low beam attenuation was used.

To eliminate or attenuate this saturation effect, a device is used whoseprinciple is indicated in FIG. 66, in which the SLM is represented as ifit functioned in transmission, and in which the polarizer associatedwith the SLM has not been represented.

The wave coming from the object and having traversed the objective isfiltered in an image plane by the diaphragm (4700). A lens (4703) makesit possible to form, in its rear focal plane, which constitutes afrequency plane, a frequency image of this wave. In the precedingembodiment, a CCD was placed directly in this frequency plane. In thepresent embodiment, an SLM (4704) is placed there. The direct beam(4702), not deflected by the sample, is represented by a broken line.Its image on the SLM is a point image. By darkening the correspondingpixel, and possibly some near pixels, this point of intense illuminationis eliminated. The other pixels of the SLM are left in the passingposition, thus enabling a ray (4701) of another frequency to passthrough the SLM.

However, the SLM is not a “perfect” system in that, in the zone in whichit is left transparent, it in fact constitutes a “grid”, each pixelbeing passing but a certain darkened space being left between twopixels. This grid diffracts the rays passing through it, generatingundesirable diffracted rays which are superimposed on the useful beam.The broken line (4710) shows a possible direction of this beam at theexit of the SLM (4704). A lens (4705) makes it possible, from the beamhaving traversed (4704), to reform a spatial plane identical to the onein which is placed (4700). In this spatial plane, undesirable raysdiffracted by (4704) are outside the image of the diaphragm (4700). Adiaphragm (4706) placed in a spatial plane and whose aperture coincideswith the image of the aperture of the diaphragm (4700) thus enablesthese diffracted rays to be eliminated.

A last lens (4708) is used to reform a frequency plane in which isplaced the CCD (4709).

8.1.4. Obtaining a regular sampling

In the third embodiment, the values of ni,nj,nk obtained after Operation1 of Step 3 of the imaging procedure described in 7.17.2 are notintegers. As integers are necessary in the rest of the algorithm, thenearest integer is taken for each of these values. Nevertheless, thisconstitutes an approximation which can result in disturbances on thethree-dimensional image generated. In the present embodiment, theoptical system is designed so that said values of ni and nj arepractically integers, i.e. so that there is regular sampling along theaxes ni and nj. The sampling along nk remains non-regular; nevertheless,this method reduces disturbances significantly.

A given illuminating wave produced by the beam deflection systemproduces, on the one hand, a direct beam striking one of the sensorsand, on the other, an opposite indicator beam striking the other sensor.Conversely, a given pixel of a sensor may be reached by a direct beamproduced by an illuminating wave or by an opposite indicator beamproduced by another illuminating wave.

For regular sampling, each illuminating wave used must produce a directbeam and an opposite beam each reaching the center of a correspondingpixel of the corresponding CCD, the coordinates of the pixel reached bythe direct beam on a CCD being the same as those of the pixel reached bythe opposite indicator beam on the other CCD. To each pixel of the CCDlocated in the zone delimited by the aperture of the objectives mustcorrespond two illuminating beams for which the pixel is reachedrespectively by the direct beam and the opposite indicator beam.

The frequency representation obtained on the CCD can be transformed invarious ways owing to the lack of precision in the characteristics ofthe system:

By translation. This translation may be compensated by correspondingdisplacement of mirrors.

By homothetic transformation. A variation in the focal length of thelens forming the image on the CCD results in a homothetic transformationon this image.

By rotation. Part of the system comprising the two objectives,consisting of the assembly (4460) of FIG. 62, shown in detail in FIG.63, if it is not perfectly constructed, will result in the rotation ofthe image produced on the CCD.

Homothetic transformation applied to the image produced on the CCDinvalidates the exact correspondence between a pixel of the CCD, whichis fixed, and the points of impact of the direct or opposite beam, whichare modified by homothetic transformation. To prevent such atransformation, it is necessary to control accurately the focal lengthof the lens forming the image. An appropriate system allows theadjustment of this focal length.

The rotation produced by (4460) is applied only to the direct beam. Itcan be compensated by a corresponding rotation of the CCD. However, thisoperation offsets the pixels of the CCD in relation to the points ofimpact of the opposite beam. It is thus necessary to produce acorresponding rotation of the opposite indicator beam to cancel thisdeviation. An appropriate system allows this rotation.

8.1.4.1. Focal length adjustment

To obtain an optical element whose focal length is adjusted withprecision around a central value f_(c), we associate two lenses of focallength f separated by a distance d.

The focal length of the assembly is then$f_{c} = {\frac{f}{2}\quad \left( \quad \frac{1}{1 - \frac{d}{2f}} \right)}$

If we wish to adjust f_(c) over a width of +−1%, or r=0.01, then it isnecessary to have $\frac{d}{2f} = r$

 with roughly f_(c)=f/2. The value of d adopted is thus: d=4f_(c)r

The value of f is then f=f_(c)+{square root over (f_(c)(f_(c)−d))}

An assembly of focal length f_(c) adjustable to +−r is thus made up oftwo lenses of focal length f separated by a distance d with:

d=4f _(c) r

f=f _(c) +{square root over (f_(c)(f_(c)−d))}

The focal length of the assembly is adjusted by varying the distance d.

Such doublets are used at various points of the device for similarreasons.

8.1.4.2. Rotation adjustment

To adjust in rotation an illuminating beam, one uses a device describedin FIG. 69 inserted on the path of the illuminating beam in a zone inwhich this beam is parallel and hence defined by its wave vector. Thisdevice is made up of an assembly of mirrors (4901) to (4906). Themirrors (4901) (4902) are fixed. The mirrors (4903) (4904) (4905) (4906)are integral with each other and the assembly (4910) consisting of thesemirrors is mobile in rotation around an axis (4909). The arrows in theplane of the figure represent the wave vectors of the beam at each pointof the device.

The transformation of a wave vector by a mirror comprises a vectorialsymmetrization in relation to an axis orthogonal to the plane of themirror and a reversal of the direction of the vector. As the number ofmirrors is even, the reversals cancel each other and we are interestedhere in the symmetrization part. The pair of mirrors (4901) (4902)produce two successive symmetrizations with axes orthogonal to eachother, which is equivalent to a single symmetry of axis (4907).Likewise, the pair of mirrors (4903) (4904) produces a symmetry of axis(4908). The mirrors (4905) (4906) produce two vectorial symmetries withthe same axis, which cancel each other out. The operation carried out bythe entire device is thus the composite of a vectorial symmetry of axis(4907) and a vectorial symmetry of axis (4908). FIG. 70 represents, seenfrom A, the axes (4907) and (4908). When no rotation of (4910) isproduced, these axes coincide and the composite of the two symmetries isthe identity. The wave vector of the beam is not modified by the device.

When a rotation of angle a is applied to the assembly (4910), the twoaxes are offset by an angle α as indicated in FIG. 70. The compositionof the two symmetries is then a vectorial rotation of angle 2α.

The system thus makes it possible to apply to an illuminating wave avectorial rotation compensating that due to the assembly (4460).

8.2. Material description

A general diagram of the system is shown in FIGS. 61, 62, 63. In thesefigures, the elements directly equivalent to corresponding elements ofFIGS. 27 and 28 are numbered using the number of the correspondingelement of FIGS. 27 and 28 and replacing the first two digits by 43. Forexample, 2204 gives 4304. The elements not having direct equivalents inFIGS. 27 and 28 have numbers beginning with 44. The plane of FIGS. 61and 62 is a horizontal plane, the figures constituting a top view. Theelements of the system are fixed on an optical table suitably insulatedfrom vibrations. FIG. 63 represents, in several views, the part of themicroscope containing the objectives, which constitutes athree-dimensional structure.

A laser (4300) polarized in the vertical direction generates a beamwhose electric field vector is thus directed along an axis orthogonal tothe plane of the figure. This beam passes through a beam expander(4301). The beam coming from the expander is then divided into areference beam and an illuminating beam by a semi-transparent mirror(4302).

The illuminating beam passes through a diaphragm (4348), a filter (4303)allowing its intensity adjustment, then a phase shift device (4304) anda beam attenuation device (4305).

It then passes through a lens (4401). This lens focuses the beam in aplane in which there is a pinhole (4402) sufficiently large so as not todisturb the beam, which is designed to stop in part the reflected beamreturning in the opposite direction to the laser. The beam is thenreflected on to one side of a double mirror (4403). The two reflectingsides of (4403) form a right angle. The beam then passes through a lens(4404) whose front focal plane coincides with the rear focal plane of(4401), and is then directed to a phase SLM (4405). The phase SLM (4405)is placed at the rear focal point (4404). The beam reflected by (4405)passes again through (4404) and is reflected by the second side of thedouble mirror (4403). The beam then passes through a diaphragm (4406)placed at the rear focal point (4404) for the beam reflected by (4405),which will be called second rear focal point of (4404). It then passesthrough a lens (4407) whose front focal point coincides with the rearfocal point of (4404). The beam then passes through a polarizer (4408).It is then reflected on to a mirror (4409) placed slightly behind therear focal plane of (4407). It then passes through a lens doublet(4411)(4410) of the type described in 8.1.4.1. The front focal plane ofthis doublet coincides with the rear focal plane of (4407). Theamplitude SLM (4412) is placed in the rear focal plane of this doublet.The beam having passed through this doublet is directed towards (4412)which reflects it. It then passes again through the doublet, through apolarizer (4413) and a diaphragm (4414) placed in the second rear focalplane of the doublet (4410)(4411). It then passes through a doubledformed by (4415) and (4416). The front focal plane of doublet(4415)(4416) coincides with the rear focal plane of the doublet(4410)(4411). It is then directed to the amplitude SLM (4417) whichreflects it. It passes again through the doublet (4415)(4416). It isreflected by a mirror (4418), passes through a diaphragm (4419) placedat the second rear focal point of the doublet (4415)(4416), then througha polarizer (4420) and a lens (4421) whose front focal plane coincideswith the second rear focal plane of the doublet (4415)(4416). It thenreaches a semi-reflecting mirror (4307) which separates it into a rightilluminating beam FED and a left illuminating beam FEG.

The beam FEG is then reflected by a mirror (4432) and passes through alens doublet (4433)(4434) which can, depending on dimensionalconditions, be located before or after the mirror (4432). The frontfocal plane of the doublet (4433)(4434) coincides with the rear focalplane of (4421). The beam is then reflected by a mirror (4435) and thenby an assembly (4436) equivalent to the assembly (4910) of FIG. 69,mobile around an axis (4450) and made up of mirrors(4446)(4447)(4448)(4449). The beam then passes through a beamextinguisher (4437). This beam extinguisher is designed like the beamattenuator described in 7.2.2. but with a zero angle θ. The beam thenpasses through a polarization rotator (4341) and is split by thesemi-transparent mirror (4325) into a main illuminating beam directedtowards (4324), which will again be denoted FEG, and an oppositeindicator beam directed towards (4342), which will be denoted FEGI. Thedoublet (4433)(4434) thus has two rear focal planes, one in thedirection of the main beam and the other in the direction of theopposite indicator beam.

The lens (4324) is placed ahead of the rear focal plane of the doublet(4433)(4434) so that the rear focal plane of this doublet in thedirection of the main beam remains virtual. The lens (4324) forms animage of this focal plane, and this image must be in the plane of thediaphragm (4323).

The rear focal plane of this doublet in the direction of the beam FEGIcoincides with the front focal plane of a lens (4342). The beam FEGIpasses through this lens which focuses it on a mirror (4343) which mayoptionally be blocked by a shutter (4359). The beam reflected by thismirror passes through lens (4342) again and is again reflected by(4325). The beam FEGI then passes through a polarization rotator (4326),and then a polarizer (4438). It is then reflected by one side of thedouble mirror (4439). It then passes through the doublet (4440)(4441)and is directed towards the amplitude SLM (4442). The front focal planeof the doublet (4440)(4441) must coincide with the rear focal plane ofthe doublet (4433)(4434). The amplitude SLM (4442) is placed in the rearfocal plane of the doublet (4440)(4441). The beam reflected by the SLM(4442) passes again through the doublet (4440)(4441), is reflected bythe second side of (4439), and is directed towards a diaphragm (4443)placed in the second rear focal plane of the doublet (4440)(4441). Thebeam passes through (4443), then a doublet (4444)(4445), a polarizer(4353), and reaches the CCD (4329) mounted on the camera (4330). Thefront focal plane of (4444)(4445) coincides with the second rear focalplane of the doublet (4440)(4441). The CCD (4329) is placed in the rearfocal plane of (4444)(4445).

The beam FEG passes through the lens (4324) and the diaphragm (4323). Itis then reflected successively by the mirrors (4322)(4451)(4452)(4453).It passes through the objective (4319), then the sample (4318), and theobjective (4317). It is then reflected successively by the mirrors(4454)(4455)(4456)(43 14) and reaches the diaphragm (4313). Thediaphragm (4323) must be placed in the plane in which the objective(4319) normally forms the image of the sample, i.e. 160 mm from theshoulder of the objective for a standard objective. The diaphragm (4313)must be placed in the plane in which the objective (4317) normally formsthe image of the sample.

The beam FEG then passes through the lens (4312) which is placed suchthat, if a transparent plate is used (absence of disturbances by theobject), and at the exit of this lens, the beam is parallel. The beamthen passes through the polarization rotator (4338), the polarizer(4423), is reflected on one side of (4424), passes through the doublet(4425)(4426), is reflected on the amplitude SLM (4427), passes againthrough the doublet (4425)(4426), is reflected on the second side of(4424), passes through the diaphragm (4428), the doublet (4429)(4430),the polarizer (4352), and reaches the CCD (4339) mounted on the camera(4384). The image of diaphragm (4313) through the lens (4312) coincideswith the front focal plane of the doublet (4425)(4426). The SLM (4427)is in the rear focal plane of the doublet (4425)(4426). The diaphragm(4428) is in the second rear focal plane of the doublet (4425)(4426).The front focal plane of the doublet (4430)(4429) coincides with thesecond rear focal plane of the doublet (4425)(4426). The CCD (4339) isplaced in the rear focal plane of the doublet (4430)(4429).

The right illuminating beam FED passes through a lens (4431) and isreflected by a mirror (4308). Depending on dimensional conditions, theposition of the lens and of the mirror may be reversed. The beam FEDthen passes through the beam extinguisher (4422) identical to (4437),then the polarization rotator (4310). It is split by a semi-transparentmirror (4311) into a main illuminating beam which again will be denotedFED and an opposite indicator beam which will be denoted FEDI.

The beam FEDI then passes through the lens (4331), is reflected by themirror (4332), passes again through (4331), is reflected in thedirection of (4338) by the semi-transparent mirror (4311). (4332) canoptionally be blocked by a shutter (4358). (4332) is in a focal plane of(4331), and the other focal plane of (4331) coincides with the image of(4313) through (4312). The beam FEDI then follows between (4311) and(4339) a path symmetrical with the one followed by beam FEGI between(4325) and (4329).

The main illuminating beam FED follows between (4311) and (4329) a pathsymmetrical with the one followed by the main illuminating beam FEGbetween (4325) and (4339).

The reference beam, separated from the illuminating beam by thepartially transparent mirror (4302), is split into a right referencebeam FRD and a left reference beam FRG by the semi-transparent mirror(4335).

The right reference beam FRD is then reflected by the mirror (4344), andthen passes through the filter (4356) and the diaphragm (4349). It isthen split by the semi-transparent mirror (4345) into a reference beaudirected towards the CCD (4339), which will again be denoted FERD, andan opposite indicator beam which will be denoted FRDI. The beam FRDIpasses through the lens (4346), is focussed on the mirror (4347) whichreflects it, passes again through the lens (4346) and is partiallyreflected in the direction of (4430). The shutter (4357) optionallyallows this opposite indicator beam to be eliminated.

The left reference beam FRG is reflected by the mirrors (4354)(4336) andthen passes through a filter (4355), a phase shift device (4351), and adiaphragm (4350). It is then split by the semi-transparent mirror (4328)into a reference beam directed towards (4329), which will again bedenoted FRG, and an opposite indicator beam which will be denoted FRGI.The beam FRGI passes through the lens (4381), is focussed on the mirror(4382) which reflects it, again passes through (4381) in the oppositedirection, and is partially reflected by (4328). The shutter (4360)optionally enables this opposite indicator beam to be eliminated.

To help in the understanding of the diagram, the illuminating beam hasbeen represented with a solid line. It passes alternately through thefrequency planes and the spatial planes, in the sense defined in 8.1.1.In a frequency plane, the beam is concentrated at a point. In a spatialplane, it is parallel and “centered”, in that it illuminates a circularzone symmetrical in relation to the optical axis, not depending on itsdirection. The letter (E) added to the number of an element means thatthis element is in a spatial plane. In the absence of an opticalelement, the letter (E) alone can also designate a spatial plane.Similarly, the letter (F) designates a frequency plane. A letter (E) hasbeen placed on the diaphragm (4313), even though this diaphragm is notexactly a spatial plane: it is the image of this diaphragm through thelens (4312) which is a virtual spatial plane and which must beunderstood as being designated by the letter E. Likewise, the diaphragm(4323) does not correspond exactly to a spatial plane.

On the other hand, the reference beam has been represented with a brokenline. The reference beam is concentrated at a point in the spatialplanes. It is parallel and centered in the frequency planes.

The spatial and frequency planes alternate on the path of the beam. Aspatial plane and a frequency plane in succession are always separatedby a lens or a doublet. Their succession follows the logic described in8.1. A spatial plane and a frequency plane separated by a lens (or adoublet) always occupy two focal planes of this lens (or of thisdoublet).

The beam deflection device whose principle was described in 8.1.2. ismade of elements corresponding to those of FIG. 65. The SLMs (4612) and(4614) are materialized respectively by the SLMs (4405) and (4412). Thediaphragm (4615) is materialized by (4406). The device was adapted totake into account the fact that the SLMs work in reflection, to includethe polarizers, and to position the diaphragm (4406) in a planedifferent from the SLM (4412). The SLM (4417) was added for additionalfiltering of the illuminating wave, thereby improving the elimination ofspurious frequencies.

The direct wave elimination device described in 8.1.3. is used by theelements corresponding to those of FIG. 66. The SLM (4704) correspondsto the SLM (4427). The diaphragms (4700) and (4706) correspondrespectively to (4313) and (4428). The CCD (4709) corresponds to (4339).The lens (4703) corresponds to the doublet (4425)(4426). The lens (4705)corresponds to the same double traversed in the opposite direction. Thelens (4708) corresponds to the doublet (4430)(4429). Symmetricalcorrespondences are valid for the symmetrical part of the microscope.

The doublets used allow the application of the principle described in8.1.4.1. They consist of two lenses which may be moved jointly, one ofthese lenses also being movable in relation to the other.

The system formed by (4436) applies the principle described in 8.1.4.2.

On each polarizer, the passing axis is indicated by a line, representingan axis in the plane of the=figure, or a circle, representing an axis ina plane orthogonal to the plane of the figure.

On each SLM, a coordinate system is represented which constitutes theone in which the coordinates of the pixels are evaluated. On eachamplitude SLM has also been represented the position of the neutral axiswhich corresponds to beam extinction, with the same convention as forthe passing axis of the polarizers. The other possible position of theneutral axis is obtained from the extinction position by a rotation ofabout 40 degrees in one direction or the other. On the phase SLM (4405)the two positions of the neutral axis are symmetrical in relation to thevertical axis.

On the polarization rotators has been indicated a coordinate system. Aposition of the neutral axis, corresponding to an applied voltage of−5V, is the horizontal axis of the coordinate system. In the otherposition, the neutral axis is directed approximately along a vectorhaving equal coordinates on the two axes. the polarizers(4408)(4413)(4420) may be Glan-Thomson prisms, which offer the advantageof low absorption. They however entail too much spherical aberration tobe used on the path of the wave coming from the object. The polarizers(4423) (4352) (4438) (4353) are preferably dichroic polarizersconsisting of a dichroic film maintained between two sufficiently thinglass plates.

The objectives used may be conventional objectives or the objectivesdescribed in Paragraphs 7.19 to 7.21. The <<preferred embodiment>>consists in using objectives of the type described in 7.20., which willallow the improvement of the working distance and/or the aperture.

A set of lenses used in the system are achromats or composite lensesminimizing spherical aberration.

Most of the elements are mounted on positioners allowing preciseadjustment of their position. The characteristics of these positionerswill be indicated in 8.5. along with the adjustment procedure. Thesample, whose positioning characteristics are not explained in 8.5., ismounted on a three-axis translation positioner.

8.3. Dimensioning

To dimension the system, the focal distance of each lens and theaperture of each diaphragm must be indicated. For the doublets, thefocal length of the doublet will be indicated.

f_(i) designates the focal length of the lens (or doublet) number i, thelenses being numbered as follows:

Lens number in the diagram or, in parentheses, Index i number of the twolens forming a doublet 1 4401 2 4404 3 4407 4 (4410,4411) 5 (4415,4416)6 4421 7 4431 or (4433,4434) 8 (4425,4426) or (4440,4441) 9 (4430,4429)or (4444,4445) 10 4312 or 4324 11 4331 or 4342 12 4346 or 4381

l_(i) designates the width of diaphragm number i, the diaphragms beingnumbered as follows:

Index i Diaphragm number in diagram 0 4349 or 4350 1 4348 2 4406 3 44144 4419 5 4313 or 4323 6 4428 or 4443

In addition, the following notations are adopted:

p_(c) distance between centers of two adjacent pixels, on the CCDsensors

p_(s) distance between centers of two adjacent pixels, on the phase SLM

p_(f) distance between centers of two adjacent pixels, on the amplitudeSLM

N_(pix) number of pixels on a CCD sensor or on an amplitude SLM (thesenumbers are equal).

N_(s) number of pixels on the phase SLM. Ideally we should haveN_(s)=2N_(pix) but the SLMs are not available in all sizes and we can dowith N_(s)=N_(pix) by applying an additional attenuation of theilluminating beam.

o: numerical aperture of a microscope objective

g: magnification of a microscope objective

f_(o): focal length of a microscope objective.

d_(o): distance between lens (4312) and diaphragm (4313).

The width of the reference beam must be at least: l₀=p_(c)N_(pix)

The illumination width on (4405) is:${{p_{s}\quad N_{s}} = {\frac{f_{2}}{f_{1}}\quad l_{1}}},$

whence we can write:$\frac{f_{2}}{f_{1}} = \frac{p_{s}\quad N_{s}}{l_{1}}$

To avoid unnecessary loss of power, it is preferable to have l₀=l₁ andhence$\frac{f_{2}}{f_{1}} = \frac{p_{s}\quad N_{s}}{p_{c}\quad N_{pix}}$

The width of the diaphragm (4406), which allows the passage of half thefrequencies coming from the SLM (4405) under an angle α_(max) is: l₂=f₂sin α_(max) or, with the value of sin α_(max) which results from 8.1.2.,$l_{2} = {f_{2}\quad \frac{\lambda}{2\quad p_{s}}}$

The useful part of the SLM (4412) must be the image of the diaphragm(4406), which means that:${\frac{f_{4}}{f_{3}}\quad l_{2}} = {p_{f}\quad N_{pix}}$

Based on the two preceding equations we obtain:${\frac{f_{4}}{f_{3}}\quad f_{2}} = {\frac{2p_{s}}{\lambda}\quad p_{f}\quad N_{pix}}$

The wave coming from (4412) exhibits a maximum angle${\sin \quad \beta_{\max}} = {\frac{\lambda}{2p_{f}}.}$

 The diaphragm (4414) must allow the passage of all these waves andconsequently checks:$l_{3} = {{2\quad \sin \quad \beta_{\max}\quad f_{4}\quad {or}\quad l_{3}} = {\frac{\lambda}{p_{f}}\quad {f_{4}.}}}$

The width of the diaphragm (4419) is equal to that of the diaphragm(4413), i.e. l₄=l₃

Its width is transformed into the width of the object diaphragm (4312)by: $l_{5} = {\frac{f_{7}}{f_{6}}\quad l_{3}}$

The width of the object diaphragm (4312) is equal to:$l_{5} = {\frac{\lambda}{2}\quad \frac{g}{o}\quad N_{pix}}$

From the preceding equations we can write:${f_{4}\quad \frac{f_{7}}{f_{6}}} = {\frac{1}{2}\quad \frac{g}{o}\quad p_{f}\quad N_{pix}}$

The size of the image on the SLM (4427) must be the same as on the SLM(4417), hence: ${\frac{f_{6}}{f_{5}}\quad \frac{f_{8}}{f_{7}}} = 1$

The image size on the camera is related to that on the SLM (4427) by:${p_{c}\quad N_{pix}} = {{\frac{f_{9}}{f_{8}}\quad p_{f}\quad N_{pix}\quad {{or}:\frac{f_{9}}{f_{8}}}} = \frac{p_{c}}{p_{f}}}$

The SLMs (4412) and (4417) having the same characteristics, we have:f₄=f₅

The lens (4312) must have an front focal plane coinciding with the rearfocal plane of the microscope objective, and is at a distance d_(o) fromthe image of the objective. We check that this results in:

f ₁₀ =gf _(o) +d _(o)

The focal length of (4331) or of (4346) must be sufficient to preventspherical aberration.

Any set of values complying with the above equations may in principle besuitable. A particularly simple solution consists in writing:$f = {\frac{p_{f}\quad N_{pix}}{2}\quad \frac{g}{o}}$

and imposing: f₂ = f₄ = f₅ = f₆ = f₇ = f₈ = f₁₁ = f₁₂ = f

One then easily obtains the other focal lengths: $\begin{matrix}{f_{3} = \quad {\frac{g}{o}\quad \frac{\lambda}{4\quad p_{s}}\quad f}} \\{f_{1} = \quad {\frac{p_{c}\quad N_{pix}}{p_{s}\quad N_{s}}\quad f}} \\{f_{9} = \quad {\frac{p_{c}}{p_{f}}\quad f}} \\{f_{10} = \quad {{gf}_{o} + d_{o}}}\end{matrix}$

and the diaphragm apertures: $\begin{matrix}{l_{0} = \quad {l_{5} = {l_{3} = {l_{4} = {\frac{\lambda}{2}\quad \frac{g}{o}\quad N_{pix}}}}}} \\{l_{2} = \quad {f\quad \frac{\lambda}{2p_{s}}}} \\{l_{0} = \quad {l_{1} = {p_{c}\quad N_{pix}}}}\end{matrix}$

The width of the beam at the exit of the beam expander (4301) must beslightly greater than l₀, the limitation being provided later by thediaphragms.

8.4. Operating mode

The operating mode is essentially the same as in the third embodiment.It is thus described in Paragraphs 7.4. to 7.17. and by the variantsdescribed in 7.18. If objectives such as those described in 7.20 or 7.21are used, the modifications described in these paragraphs must beapplied. The <<preferred embodiment>> must be understood as using thealgorithms described in Paragraphs 7.4. to 7.17. without the variantsdescribed in 7.8. However, certain solutions described in the variantsoffer specific advantages and the qualifier <<preferred embodiment>>must not be understood in too absolute a manner. A certain number ofdifferences must however be taken into account in relation to thealgorithms described in the third embodiment:

8.4.1. Beam deflection control

The state of SLM (4405) is given by a control array A[k,l] in in which kand l vary from 0 to N_(s)−1, the element A[k,l] corresponding to thepixel of coordinates k,l and having the value 0 for a negative phasepixel and the value 1 for a positive phase pixel. The state of the SLMs(4412) and (4417) is given by a control array B[k,l] in which k and lvary from 0 to N_(pix)−1, the element B[k,l] corresponding to the pixelof coordinates k,l and having the value 0 for an extinguished pixel andthe value 1 for a lit pixel.

To obtain a frequency characterized by the indices (i,j):

we light the pixel of coordinates i,j on the SLMs (4412) and (4417),i.e. we use an array B_(ij) where B_(ij)[k,l]=0 at every point except ati,j where we have B_(ij)[i,j]=1

we apply to the SLM (4405) an appropriate array A_(ij), namely:${A_{ij}\left\lbrack {k,l} \right\rbrack} = {E\quad \left( {\left\{ {\frac{1}{N_{pix}}\quad \left( {{k\quad \left( {i + 1} \right)} + {l\quad \left( {j + 1} \right)}} \right)} \right\} \% \quad 2} \right)}$

 in which E designates the rounding to the nearest integer and %2 meansmodulo 2.

The “control word” used in the third embodiment thus consists here ofthe concatenation of the control arrays of the SLMs (4405)(4412)(4417)and the control bits of the phase rotators of (4422) and (4437). Aspreviously, the directly illuminated sensor is designated by the index pwith p=0 for (4339) and p=1 for (4329). For the control of the phaserotators included in the beam extinguishers (4422) and (4437), a zerocontrol bit corresponds to an applied voltage of 5V (open position) anda control bit at 1 corresponds to an applied voltage of −5V (closedposition).

Control word COM[p, i, j] Control array of SLM (4405)${A_{ij}\left\lbrack {k,l} \right\rbrack} = {E\quad \left( {\left\{ {\frac{1}{N_{pix}}\quad \left( {{k\left( {i + 1} \right)} + {l\left( {j + 1} \right)}} \right)} \right\} \quad {\% 2}} \right)}$

Control array of SLM (4412) B_(ij) [i, j] = 1 B_(ij) [k, l] = 0 if (k,l) ≠ (i, j) Control array of SLM (4417) B_(ij) [i, j] = 1 B_(ij) [k, l]= 0 if (k, l) ≠ (i, j) Control bits of rotators of (4422) {overscore(p)}, {overscore (p)} Control bits of rotators of (4437) p, p

The control word thus formed is substituted in all the procedures of7.4. to 7.17. for the one which was formed as indicated in 7.2.4.

8.4.2. Use of the direct wave elimination system

The SLMs (4427) and (4442) are respectively associated with the sensors(4339) and (4329) and indexed by the same indices p=0 and p=1. In theprocedure described in 7.12.2.1., Phase 1, during the acquisition of apair of elementary images, it is necessary to also control these SLMs.The value of an element C[k,l] of the control array used for such an SLMdepends on the index of the SLM, the indices c and p of the image beingacquired, and the indices i and j corresponding to the point of thesensor illuminated directly by the illuminating beam, ori=Ia[q,p,Ic[k,p],Jc[k,p]],j=Ja[q,p,Ic[k,p],Jc[k,p]]. It is given by thefollowing table in which r designates an “extinction radius” which maybe taken equal to 2 for example.

Index c 2 Other (non-maximum attenuation) Index of SLM {overscore (p)} pIndices k, l (k − i)² + (l − j)² > r² (k − i)² + (l − j)² ≦ r² Value ofelement of array C[k, l] 1 1 1 0

8.4.3. Use of regular sampling

8.4.3.1. Modification of procedure 7.9.2.

The coordinates of the illuminating beams are known in advance and are:

Ia[q,p,i,j]=i{overscore (p)}+(N _(pix) −i−1)p

Ja[q,p,i,j]=j

It is thus sufficient to determine the array Ra[p,i,j]

We use the arrays Io and Jo characterizing a path going through all theaccessible points, i.e. (Io[k], Jo[k]) must go through all the valuessuch that${\left( {{{Io}\lbrack k\rbrack} - \frac{N_{pix}}{2}} \right)^{2} + \left( {{{Jo}\lbrack k\rbrack} - \frac{N_{pix}}{2}} \right)^{2}} \leq R_{ouv}^{2}$

where R_(ouv) is the radius of the disc limited by the aperture of theobjective on a sensor, and corresponds for example to the radius of theilluminated zone on (4339) when the beam FRGI is used alone. This pathwill be called “complete path” hereafter.

A program performs the acquisition defined by these arrays according tothe procedure described in 7.12. As the index n_(o) is not known, it istaken equal to n_(v) in the procedure 7.12. However, during thisacquisition, it is sufficient to record the values M_(k,p,q)[Jo[k],Jo[k]] and H_(k,p,q)[i_(r),j_(r)].

The program initializes to 0 the array Ra and then goes through theseries of indices k,p performing, for each pair k,p:${{Ra}\left\lbrack {p,{{Io}\lbrack k\rbrack},{{Jo}\lbrack k\rbrack}} \right\rbrack} = {\frac{M_{k,p,0}\left\lbrack {{{Io}\lbrack k\rbrack},{{Jo}\lbrack k\rbrack}} \right\rbrack}{H_{k,p,0}\left\lbrack {i_{r},j_{r}} \right\rbrack}\quad \exp \quad \left( {{- \overset{\sim}{j}}\quad \frac{2\quad \pi}{\lambda}\quad n_{v}\quad \left( {{x\quad \frac{{Io}\lbrack k\rbrack}{K}} + {y\quad \frac{{Jo}\lbrack k\rbrack}{K}} + {z\quad \sqrt{1 - \frac{{{Io}\lbrack k\rbrack}^{2} + {{Jo}\lbrack k\rbrack}^{2}}{K}}}} \right)} \right)}$

where i_(r),j_(r) are the coordinates of the maximum of the referenceimage, as defined in 7.12. and where x,y,z are the coordinatesdetermined in 7.9.1.

8.4.3.2. Modification of procedure 7.11.

The program carries out the series of acquisitions defined by the arraysIo and Jo defining a complete path, already used in 8.4.3.1., accordingto the procedure described in 7.12. It thus generates the series ofimages M_(k,p,q)[i,j] and H_(k,p,q)[i,j]. However, during thisacquisition, it is sufficient to record the value M_(k,p,q)[Io[k],Jo[k]] and H_(k,p,q)[i_(r),j_(r)].

The program then goes through the series of indices k. For each value ofk it carries out:${F_{rec}\left\lbrack {{{Io}\lbrack k\rbrack},{{Jo}\lbrack k\rbrack}} \right\rbrack} = {\frac{M_{k,0,0}\left\lbrack {{{Io}\lbrack k\rbrack},{{Jo}\lbrack k\rbrack}} \right\rbrack}{H_{k,0,0}\left\lbrack {i_{r},j_{r}} \right\rbrack}\quad \frac{1}{{Ra}\left\lbrack {0,{{Io}\lbrack k\rbrack},{{Jo}\lbrack k\rbrack}} \right\rbrack}}$

The program described in 7.8. is then used to calculate the parametersx,y,z,L, n₀ from the array F_(rec) thus formed.

The procedure 7.11. thus reformulated may be used directly to calculatex,y,z,L,n₀ in the case of the uniaxial crystal described in 7.18.9.,thereby obviating the need to perform preliminary measurements andmaking the movement of the objectives possible. In this case, theacquisition method for the series of images allowing the calculation ofx,y,z,L,n₀ corresponds to the procedure described in 7.12 and modifiedas indicated in 7.18.9.

8.4.3.3. Modification of procedure 7.13.

In 7.13, one obtains directly without using the program of FIG. 53:

Id[p,i,j]=i{overscore (p)}+(N _(pix) −i−1)p

Jd[p,i,j]=j

Ic[k,p]={overscore (p)}Io[k]+(N _(pix) −Io[k]−1)p

Jc[k,p]=Jo[k]

8.4.3.4. Modification of procedure 7.6.

The values of the coefficients K₁, K₂ determined in 7.6. comply with:K₁=K₂

8.5. Adjustment

The position of each element of the system must be adjusted preciselybefore any use.

8.5.1. Apparatus used

The apparatus already described in 7.3.3.2. are used.

8.5.2. Types of images used

During the adjustment, different types of images may be used:

images obtained on an auxiliary CCD: an auxiliary CCD placed for examplein a spatial plane can allow the determination of the center of anilluminating beam in this plane, or the punctuality of a reference beamin this plane.

images obtained on one of the CCDs of the microscope: these images maybe obtained and analyzed as indicated in 7.3.3.1. in the presence of areference beam. It is also possible to observe directly the imagesreceived in the absence of a reference beam.

images obtained on the CCD of the frequency-meter: observed directly,they make it possible to check the flatness of a wave or the anglebetween two plane waves.

images of the surface of an SLM: by placing the frequency-meter behindthe lens which transforms into a plane wave the wave coming from a givenpoint of the SLM, one forms on the CCD of the frequency meter an imageof the surface of the SLM, which can be used for example to check thatthe SLM is correctly illuminated. To display the figures formed bycontrolling the SLM, a polarizer must moreover be present between theSLM and the CCD of the frequency-meter. If it is not already present inthe system, it is possible to use the polarizer of the frequency-meter.

pixel-by-pixel images of the surface of an SLM: such an image comprisesan array of two dimensions containing the illumination of each pixel. Toobtain it, the frequency-meter is placed as above in the presence of apolarizer. The entire SLM is placed in an absorbing position (“black”image). Then each pixel is illuminated one by one, recording each timethe intensity of the corresponding point on the frequency-meter. Theintensities thus measured for each pixel of the SLM are stored in anarray, which forms the pixel-by-pixel image of the SLM. This type ofimage can be used for example to check the point-by-point correspondencebetween the various SLMs, necessary for obtaining correct illuminationand regular sampling.

8.5.3. Adjustment criteria

The adjustments are designed to ensure that:

(1) the beams follow the planned path. This can be checked in general bymeans of a simple diffuser.

(2) the illuminating beams are parallel in the spatial planes. This isverified by means of the frequency-meter.

(3) the reference beams are punctual in the spatial planes and theilluminating beams are points in the frequency planes. This is verifiedfor example by means of an auxiliary CCD.

(4) the polarizers are properly adjusted. Beam extinctions can beobserved for example on the frequency-meter.

(5) a parallel beam entering the objective (4317) and directed along theoptical axis has a point image centered on the sensor (4339).

(6) when the control word A_(i,j) is used for the SLM (4405) and whenthe control word B_(i,j) is used for the SLMs (4412) and (4417), thefollowing conditions are complied with:

(i) the point of coordinates i,j must be actually illuminated on theSLMs (4412) and (4417)

(ii) when the beams FEG and FEGI are used, the points illuminated on theSLMs (4427) and (4442) and on the sensors (4339) and (4329) must havethe coordinates (i,j)

(iii) when the beams FED and FEDI are used, the illuminated points onthe SLMs (4427) and (4442) and on the sensors (4339) and (4329) musthave the coordinates (N_(pix)−1−i,j)

This condition (6) can be checked by means of images obtained on thesensors or by means of pixel-by-pixel images obtained on the CCDs.

The adjustments to be carried out result from these conditions. Thedescription of the adjustment steps is given by way of information andconstitutes an example of adjustment step ordering.

8.5.4. Adjustment steps

In the description of the adjustment steps, reference will frequently bemade to the optical axis. Owing to the many reflections, the opticalaxis can be defined only locally. It is thus this locally definedoptical axis to which reference will be made.

Prior to any fine adjustment, the entire system is put in place with thegreatest possible precision using geometrical methods, with theexception of the elements (4304) (4305) (4422) (4437) (4351) which willbe put in place during adjustment.

Throughout the adjustment, the position of the neutral axis of thepolarization rotators (4310) (4338) (4341) (4326) is kept in the planeof FIG. 62. A point will be said to be centered on one of the sensors(4339) or (4329) if its coordinates are$\left( {\frac{N_{pix}}{2},\frac{N_{pix}}{2}} \right).$

During the first adjustment step for a given element, the type ofpositioner on which this element is mounted is indicated.

Step 1. Orientation Adjustment for the Laser (4300)-beam Expander (4301)Assembly

This assembly is mounted on an angular positioner allowing the directionof the beam to be adjusted. A diffuser is used to check the path of thebeam. The position of the assembly (4300,4301) is adjusted so that thebeam follows the planned path.

Step 2. Setup of the Phase Shift Device (4304)

This phase shift device is identical to the one described in 7.2.3. andis put in place as indicated in 7.3.2.3.

Step 3. Setup of Beam Attenuation Device (4305)

This beam attenuation device is identical to the one described in 7.2.2.and is put in place as indicated in 7.3.2.2.

Step 4. Pinhole (4402) Two-axis Translation Adjustment

This pinhole is mounted a two-dimensional positioner allowing movementin the plane orthogonal to the optical axis.

It is adjusted so as to maximize the intensity of the beam having passedthrough the pinhole.

Step 5. Adjustment of Orientation of Two-sided Mirror (4403)

This mirror is mounted on an angular positioner allowing its orientationto be adjusted. A diffuser is used to check the path of the beam and theposition of the mirror is adjusted so that the incident beam on the SLM(4405) occupies the entire useful surface of this SLM.

Step 6. SLM (4405) Orientation Adjustment

This SLM is mounted on an angular positioner allowing its orientation tobe adjusted, coupled with a two-axis translation positioner allowing anadjustment of its position in a plane orthogonal to the optical axis.

The SLM is positioned to be totally reflecting. A diffuser is used tocheck that the wave reflected on the SLM reaches the planned point onthe mirror (4403) and is directed in the planned direction.

Step 7. Fine Adjustment of Orientation of Two-sided Mirror (4403),Translation Adjustment of the SLM (4405), Adjustment of Aperture ofDiaphragm (4348), and Adjustment of Lens (4404) in Translation

The lens (4404) is mounted on a single-axis translation positionerallowing translation adjustment in the direction of the optical axis.

This step is designed to:

adjust the position of the lens (4404) so that the SLM (4405) is in aspatial plane.

adjust the orientation of the two-sided mirror (4403) and thetranslation position of (4405) so that the beam reaching (4405) iscentered.

adjust the aperture of (4348) so that the incident beam on (4405) is aswide as possible, without however exceeding the active surface of theSLM.

It is possible, on an SLM, to control independently the active zone anda peripheral zone called the “apron.” The entire active zone here iscontrolled to perform a polarization rotation in a given direction, andthe apron is controlled to carry out a polarization rotation in theopposite direction. The active zone is then modified so that a crosscentered at the middle of the active zone is placed in the same state asthe apron.

The frequency-meter is positioned behind (4406), which is widely open.

The aperture of the diaphragm (4348) is slightly greater (20% forexample) than its nominal aperture calculated in 8.4.

The frequency-meter is first positioned in the absence of its polarizerso as to form on its CCD an image of the illuminated zone of (4405).

The polarizer of the frequency-meter is then introduced and adjusted inrotation so as to cause the appearance on the image of a maximumcontrast between the peripheral zone (apron) and the central active zoneof the SLM.

The position of (4403) is then adjusted so that the entire active zoneand the border with the apron is visible on the image. One should see anilluminated square active zone surrounded by a dark zone (the apron) andmarked by a centered cross.

The position of (4404) is adjusted so as to have the best possiblecontrast.

The aperture of the diaphragm (4348) is then reduced. We should see onthe image an illuminated disc of reduced diameter intercepting the darkcross. The position of (4403) and that of (4405) in translation areadjusted so that the center of the disc and the center of the crosscoincide.

The position of (4404) is again adjusted so as to have the best possiblecontrast.

The opening of the diaphragm (4348) is then enlarged as much as possiblewithout however having the illuminated disc reach the apron zone.

Step 8. Translation Adjustment of Lens (4407)

This lens is mounted on a single-axis translation positioner in thedirection of the optical axis.

The purpose of this adjustment is to ensure that a parallel beam comingfrom the laser and reflected entirely by (4405) is again parallel afterpassing through (4407).

The frequency-meter is positioned behind (4407). (4406) is open widely.The control array of (4405) is entirely set to 0. The position of (4407)is adjusted so as to have an image as close as possible to a point imageon the frequency-meter.

Step 9. Rotation Adjustment of Polarizer (4408)

The polarizer (4408) is attached to a positioner allowing a rotationadjustment around the optical axis.

When the control word A₀₀ is used, the SLM (4405) reflects the beam in adirection D0. When A_(N) _(pix) _(−1,0) is used, the direction of thebeam is modified and the component on the frequency corresponding to thedirection D0 must be cancelled. The polarizer (4408) is adjusted so asto actually produce a cancellation of this component.

The frequency-meter is positioned behind (4408). The diaphragm (4406) iswidely open. The array A₀₀ is first applied to (4405). The illuminatedpoint P on the CCD of the frequency-meter is marked by a coordinatesystem. The array A_(N) _(pix) _(−1,0) is then used and the position of(4408) is adjusted to cancel the intensity received at the point P.

Step 10. Two-axis Translation Adjustment and Aperture of Diaphragm(4406)

The diaphragm (4406) has an adjustable aperture and is adjustable intranslation along two axes orthogonal to the optical axis.

Its position and its aperture must be adjusted so that it allows thepassage of useful frequencies and stops the symmetrical frequencies, itsrole being that of the diaphragm (4615) described in.8.1.2.

The frequency-meter is used and positioned behind (4408). The axes ofthe CCD of the frequency meter must be oriented as indicated by thearrow marker (4470). A specific program is used.

This program applies successively to the SLM (4405) the arrays:$A_{0,0}\quad A_{0,\frac{N_{pix}}{2}}\quad A_{\frac{N_{pix}}{2},0}\quad A_{\frac{N_{pix}}{2},{N_{pix} - 1}}\quad A_{{N_{pix} - 1},\frac{N_{pix}}{2}}$

The program sums the intensities obtained in each case by the CCD sensorand displays the corresponding image. It superimposes on this imagesymbols indicating the maximum obtained in each of these cases. Ititerates indefinitely this procedure so that this image is updatedconstantly throughout the adjustment of the diaphragm.

The position and aperture of the diaphragm must be adjusted so that thepoint obtained for the indices (0,0) is not visible (blocked by thediaphragm) and so that the four other points are visible (at the limitof the diaphragm), at the positions indicated in FIG. 72 in which (5010)represents the limit of the diaphragm.

Step 11. Mirror (4409) Orientation Adjustment

The mirror (4409) is fixed on a positioner allowing its orientation tobe adjusted.

The array $A_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

is applied to (4405). A diffuser is used to control the path of thebeam. The angular position of (4409) is adjusted so that the beamreaches the center of (4412).

Step 12. SLM (4412) Orientation Adjustment

The SLM (4412) is fixed on a three-axis rotation positioner allowing theadjustment of its orientation (axes in the plane of the sensor) and itsrotation position (axis orthogonal to the plane of the sensor), coupledwith a two-axis translation positioner allowing a displacement in aplane orthogonal to the optical axis.

The array $A_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

remains applied to (4405). The control array of (4412) is set to 1. Adiffuser is used the check the path of the beam. (4412) is adjusted sothat the wave is reflected towards (4414) and is centered at the middleof this diaphragm.

Step 13. Polarizer (4413) Rotation Adjustment

The control array of (4412) is set to 1. The frequency-meter ispositioned behind (4413) so that the image of (4412) is formed on itsCCD. The control array of (4412) is then set to 0. (4413) is thenadjusted in rotation so as to cancel the intensity received on the CCDof the frequency-meter.

Step 14. Fine Adjustment of Orientation of Mirror (4409), Adjustment ofPosition and Focal Length of Doublet (4410) (4411), and Rotation andTranslation Adjustment of SLM (4412)

The lens (4410) is mounted on a single-axis positioner in the directionof the optical axis. This positioner and the lens (4411) are themselvesmounted on a second single-axis positioner in the direction of theoptical axis. It is thus possible either to move the assembly(4410)(4411) jointly or move (4410) alone to vary the distance betweenthe two lenses of the doublet (4410)(4411).

The SLM (4405) makes it possible to control the direction in which isdiffracted the wave coming from this SLM. To a direction of the beam atthe exit of the SLM (4405) there corresponds a point in a frequencyplane and in particular a point in the plane of the SLM (4412).

The position of (4410)(4411) must be adjusted so that the image of thebeam coming from the SLM (4405) is actually a point image in thefrequency plane in which (4412) is located. The position of (4409) and(4412) must be adjusted so that the frequency generated when the controlarray $A_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

is applied to (4405) corresponds to a point of coordinates$\left( \quad {\frac{N_{pix}}{2},\frac{N_{pix}}{2}} \right)$

on the SLM (4412). The position of (4410)(4411)(4412) must moreover beadjusted so that there is a point-to-point correspondence between thefrequencies generated by the arrays A_(i,j) applied to (4405) and thepixels of (4412), i.e. so that when the control array A_(i,j) is appliedto (4405) the pixel of coordinates (i,j) is illuminated on the SLM(4412), whatever the integers i and j.

The series of operations o1 to o3 is repeated a sufficient number oftimes so as to converge towards the correct position of each element.

o1. Adjustment of Joint Position of (4410) and (4411) to Obtain a PointFrequency Image

The frequency-meter is positioned behind (4414). The diaphragm (4414) isopen widely. The array $A_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

is applied to (4405) and the control array applied to (4412) is at 1.The joint position of the assembly (4410) (4411) is adjusted to have, onthe CCD of the frequency-meter, an image which is as much as possible apoint image.

o2. Angular Adjustment of (4409) and Translation Adjustment of (4412) toObtain a Centered Frequency Image$A_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

remains applied to (4405). (4409) and (4412) are then adjusted so thatthe central frequency generated is in the middle of the SLM (4412). Thisadjustment is carried out with the frequency meter and a program forlocating the maximum, which will be called program PA.

The program PA illuminates one by one each pixel of the SLM (4412) andmeasures the corresponding intensity on the frequency-meter. The pixelgenerating the highest intensity corresponds to the maximum.

The basic procedure of the program PA, which will be called PB, goesthrough all the coordinates (i,j) of the pixels on the SLM (4412), i andj varying from 0 to N_(pix)−1. For each pair (ij) this procedure carriesout the following steps:

it applies the control array B_(ij) defined as in 8.4.1. to the SLM(4412).

it then acquires an image on the CCD of the frequency-meter.

it determines the maximum value of the intensity measured on all thepoints of this image.

it records this value in an array M at M[i,j]

When it has thus gone through all the indices i,j, the proceduredetermines the point of maximum intensity of the array M and its indicesi₀,j₀ which correspond to the maximum. It displays on the screen thetwo-dimensional image corresponding to the array M, possibly magnifiedaround the maximum, as well as the values of i₀,j₀ and the valueM[i₀,j₀].

The program PA consists in iterating indefinitely the procedure PB so asto be able to carry out the corresponding adjustment.

The position of (4409) must be adjusted to have$\left( {i_{0},j_{0}} \right) = \left( \quad {\frac{N_{pix}}{2},\frac{N_{pix}}{2}} \right)$

and to maximize M[i₀,j₀].

o3. Adjustment of Focal Length of Doublet (4410)(4411) and Adjustment of(4412) in Rotation

The frequency-meter, adjusted as previously, is used. A programdisplaying the characteristics of the system is used, and will be calledprogram PC.

The program PC uses successively four control arrays$A_{c,\frac{N_{pix}}{2}}\quad A_{\frac{N_{pix}}{2},c}\quad A_{\frac{N_{pix}}{2},{N_{pix} - 1 - c}}\quad A_{{N_{pix} - 1 - c},\frac{N_{pix}}{2}}$

applied to (4405). These control arrays are numbered in that order. c isa constant, for example c=20, introduced to prevent the illuminatedpoints from leaving the active zone of the SLM in the event of incorrectinitial adjustment.

For the n-th control word, the program uses the procedure PB alreadydescribed and stores the coordinates of the maximum obtained at X[n] andY[n]. When this operation has been carried out for the four controlwords, the program has obtained the arrays X and Y with four elements,corresponding to the coordinates in pixels of the successive maximums.When the system is correctly adjusted, we should have:${\left( {{X\lbrack 1\rbrack},{Y\lbrack 1\rbrack}} \right) = \left( {c,\frac{N_{pix}}{2}} \right)}\quad$$\left( {{X\lbrack 2\rbrack},{Y\lbrack 2\rbrack}} \right) = \left( {\frac{N_{pix}}{2},c} \right)$${\left( {{X\lbrack 3\rbrack},{Y\lbrack 3\rbrack}} \right) = \left( \quad {\frac{N_{pix}}{2},{N_{pix} - 1 - c}} \right)}\quad$$\left( {{X\lbrack 4\rbrack},{Y\lbrack 4\rbrack}} \right) = \left( {{N_{pix} - 1 - c},\frac{N_{pix}}{2}} \right)$

The adjustment thus seeks to obtain these equalities effectively. Theadjustment of the focal length of the doublet (4410)(4411) allows theadjustment of the scale, and the rotation adjustment of (4412) allowsthe adjustment of the rotation position

The program calculates the ratio of the actual distances to thedistances which should be obtained with a correct adjustment:$r = {\frac{1}{N_{pix} - 1 - {2c}}\quad \frac{1}{2}\left\{ {\sqrt{\left( {{X\lbrack 1\rbrack} - {X\lbrack 4\rbrack}} \right)^{2} + \left( {{Y\lbrack 1\rbrack} - {Y\lbrack 4\rbrack}} \right)^{2}} + \sqrt{\left( {{X\lbrack 2\rbrack} - {X\lbrack 3\rbrack}} \right)^{2} + \left( {{Y\lbrack 2\rbrack} - {Y\lbrack 3\rbrack}} \right)^{2}}} \right\}}$

If f is the focal length of each lens and if d₁ is the distance betweenthese two lenses before adjustment, the actual focal length of theassembly of the two lenses before adjustment is:$f_{1} = {\frac{f}{2}\quad \left( \frac{1}{1 - \frac{d_{1}}{2f}} \right)}$

Similarly, the focal length of the assembly after adjustment is:$f_{2} = {\frac{f}{2}\quad \left( \frac{1}{1 - \frac{d_{2}}{2f}} \right)}$

As magnification is proportional to the focal length, it must beadjusted so that: $\frac{f_{2}}{f_{1}} = \frac{1}{r}$

which leads, by developing the calculations, to:${d_{2} - d_{1}} = {2f\quad \left( {1 - r} \right)\quad \left( {1 - \frac{d_{1}}{2f}} \right)}$

In this equation, f and d₁ are poorly known but the “design” focallength doublet (4410)(4411) can be used. This length is that for whichthe doublet was designed. It will be denoted f_(c) and will be equal toroughly f/2. We then obtain:

d ₂ −d ₁=4f _(c)(1−r)

The program PC displays:

the value d₂−d₁, which allows the distance between lenses to becorrected accordingly.

the lines linking respectively the points 1 and 4 and the points 2 and 3

the deviation$\frac{{X\lbrack 1\rbrack} - {X\lbrack 4\rbrack}}{N_{pix} - {2c}},$

 which gives roughly the angle in radians by which the rotation positionmust be corrected.

the ratio$\frac{{X\lbrack 1\rbrack} - {X\lbrack 4\rbrack}}{{Y\lbrack 2\rbrack} - {Y\lbrack 3\rbrack}},$

 which should be equal to ±1.

The SLM (4412) is adjusted in rotation so as to cancel the displayeddeviation$\frac{{X\lbrack 1\rbrack} - {X\lbrack 4\rbrack}}{N_{pix} - {2c}},$

and the distance between lenses of the doublet (4410)(441 1) is modifiedaccording to the displayed value of d₂−d₁.

Step 15. Translation Adjustment of Diaphragm (4414)

The diaphragm (4414) is mounted on a two-axis translation positionerallowing it to be moved in a plane orthogonal to the optical axis.

The aperture of (4414) is known. It must be adjusted in translation. Anauxiliary CCD is placed just behind (4414). The control array$B_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

is used for the SLM (4405), i.e. only a central point of this SLM is“lit”, the direct reflected beam being stopped by (4406). The controlarray of the SLM (4412) is set to 1 (passing position). The image of thelit point of (4405) is then formed in the plane of the diaphragm (4414).(4414) is then adjusted in translation so that its center coincides withthe image of the lit point of (4405).

Step 16. Two-axis Translation Adjustment of the SLM (4417)

This SLM is mounted on a two-axis translation positioner in the plane ofthe SLM, coupled with a three-axis rotation positioner for adjusting itsorientation (axes in the plane of the sensor) and its rotation position(axis in a plane orthogonal to the sensor).

The control arrays $A_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

and $B_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

are used respectively for the SLMs (4405) and (4412). A diffuser is usedto follow the beam arriving on (4417), and (4417) is adjusted intranslation so that the beam arrives at its center.

Step 17. Orientation Adjustment of the SLM (4417)

The same control arrays as previously are used for the SLMs(4405)(4412). The control array of the SLM (4417) is set to 1. (4417) isadjusted so that the reflected beam reaches the point provided for on(4419), which is verified by means of a diffuser.

Step 18. Orientation Adjustment of the Mirror (4418)

The mirror (4418) is mounted on an angular positioner allowing itsorientation to be adjusted. (4418) is adjusted so as to effectively sendback the beam in the planned direction, which is verified with adiffuser.

Step 19. Translation Adjustment of the Diaphragm (4419)

The diaphragm (4419) is mounted on a two-axis translation positionerallowing displacement in a plane orthogonal to the optical axis.

The aperture of (4419) is known. It must be adjusted in translation. Thecontrol array $B_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

is used for the SLM (4405), i.e. only a central point of this SLM is“lit”, the direct reflected beam being stopped by (4406). The controlarrays of the SLMs (4412) and (4417) are set to 1 passing position). Theimage of the lit point of (4405) is then formed in the plane of thediaphragm (4419). (4419) is then adjusted in translation so that itscenter coincides with the image of the lit point of (4405).

Step 20. Rotation Adjustment of the Polarizer (4420)

The polarizer (4420) is mounted on a positioner allowing its rotationadjustment in relation to the optical axis.

The control arrays $A_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

and $B_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

are applied respectively to (4405) and (4412). The control array of(4417) is set to 1. The frequency-meter is placed just behind (4420).The image on its CCD is roughly a point image. The control array of(4417) is then set to 0. The position of (4420) is adjusted to cancelthe intensity received on the frequency-meter.

Step 21. Two-axis Translation Adjustment and Rotation Adjustment of theSLM (4417), Adjustment of Position and Focal Length of the Doublet(4415) (4416)

The lenses (4415)(4416) are mounted on a translation positioner allowingeither their simultaneous displacement in the direction of the opticalaxis or the displacement of (4415) alone.

The position of (4415)(4416) must be adjusted so that the image of thebeam coming from the SLM (4412) when the control arrays$A_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

and $B_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

are applied respectively to (4405) and (4412) is effectively a pointimage in the frequency plane in which (4417) is located. The position of(4417) must be adjusted so that, under these conditions, the lit pointon the SLM (4417) has the coordinates$\left( \quad {\frac{N_{pix}}{2},\frac{N_{pix}}{2}} \right).$

The position of (4415)(4416)(4417) must moreover be adjusted so thatthere is a point-to-point correspondence between the pixels of (4412)and those of (4417), i.e. so that, when the control arrays A_(i,j) andB_(i,j) are applied respectively to (4405) and (4412), the lit point on(4417) is the pixel of coordinates (i,j), whatever the integers i and j.

The frequency-meter is positioned behind (4420). The series ofoperations o11 to o13 is repeated a sufficient number of times.

o11. Adjustment of Joint Position of (4415) and (4416) to Obtain a PointFrequency Image

The control arrays $A_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

and $B_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

are applied respectively to the SLMs (4405) and (4412).

The control array applied to (4417) is set to 1. The position of theassembly (4415) (4416) is adjusted to have, on the CCD sensor of thefrequency-meter, an image as close as possible to a point image.

o12. Translation Adjustment of (4417) to Obtain a Centered FrequencyImage $A_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

and $B_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

remain applied respectively to the SLMs (4405) and (4412). (4417) isthen adjusted so that the central frequency generated is in the middleof the SLM (4417). This adjustment is carried out with thefrequency-meter and with the maximum-location program PA defined above.However, in this maximum-location program, the SLM (4412) is replacedhere by the SLM (4417).

o13. Adjustment of (4415) to Obtain the Correct Frequency Magnificationand Rotation Adjustment of (4417)

The program PC is used. However, in this program:

the SLM (4412) is replaced by the SLM (4417)

when a control word A_(i,j) is applied to the SLM (4405), thecorresponding control word B_(i,j) is applied to the SLM (4412).

The inter-lens distance correction obtained is applied to the distancebetween (4415) and (4416) and the rotation adjustment is carried out on(4417) in rotation in relation to an axis orthogonal to the plane of thesensor.

Step 22. Orientation Adjustment of Semi-transparent Mirror (4307)

The semi-transparent mirror (4307) is mounted on an angular positionerallowing its orientation to be adjusted.

The SLMs (4405)(4412)(4417) are controlled to generate a centralfrequency: the control array of (4405) is$A_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

and the control array of (5512) and (4417) is$B_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}.$

The position of (4407) is adjusted by means of a diffuser in order tosend the beam back in the appropriate direction.

Step 23. Orientation Adjustment of the Mirror (4308)

The mirror (4308) is mounted on an angular positioner allowing itsorientation to be adjusted.

The control of the SLMs is unchanged. The position of (4308) is adjustedby means of a diffuser so as to send the beam back in the appropriatedirection.

Step 24. Setup of Beam Extinguisher (4422)

The control of the SLMs is unchanged. This beam extinguisher is placedin accordance with the procedure indicated in 7.3.2.2.

Step 25. Translation Adjustment of Lens (4431)

The control of the SLMs is unchanged. The frequency-meter is placedbehind (4431). The position of (4431) is adjusted to obtain a pointimage.

Step 26. Orientation Adjustment of Mirror (4332)

This mirror is mounted on a two-axis positioner enabling its orientationto be adjusted.

The control of the SLMs is unchanged. Between (4311) and (4331), twobeams propagating in opposite directions are superimposed. A diffuserintroduced on the side of the beam will be lit on both sides. The litparts of each side of the diffuser must be the same: the two beams arethen exactly superimposed. Incorrect adjustment of (4332) has the effectof offsetting the position of these two beams. (4332) is adjusted sothat between (4311) and (4331) the beams propagating in both directionsare exactly superimposed.

Step 27. Translation Adjustment of Lens (4331)

The lens (4331) is mounted on a single-axis translation positionerenabling displacement in the direction of the optical axis.

The frequency-meter is positioned between (4311) and (4338). The controlof the SLMs (4405)(4412)(4417) is defined by the arrays$A_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

and $B_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}.$

The position of (4332) is adjusted so that FEDI has an image as close aspossible on the frequency-meter.

Step 28. Orientation Adjustment of the Semi-transparent Mirror (4311)

This semi-transparent mirror is mounted on a two-axis rotation position(the two axes being in the plane of the semi-transparent mirror)enabling its orientation to be adjusted.

The adjustment of the SLMs is not modified. A diffuser is used tocontrol the path of the beam. (4311) is adjusted so that the beam FEDIfollows the appropriate path.

Step 29. Orientation Adjustment of the Two-sided Mirror (4424)

The double mirror (4424) is mounted on a two-axis rotation positioner,the two axes being in the plane of the non-reflecting side.

(4424) is adjusted so that FEDI reaches the center of (4427), which isverified by means of a diffuser.

Step 30. Orientation Adjustment of the SLM (4427)

This SLM is mounted on a two-axis translation positioner in the plane ofthe SLM, coupled with a three-axis rotation positioner allowing theadjustment of its orientation (axes in the plane of the sensor) androtation position (axis in a plane orthogonal to the sensor).

The control arrays of the SLMs (4405) (4412) (4417) are not modified.The control array of the SLM (4427) is set to 1.

With the path of the beam controlled by means of a diffuser, the angularposition of (4427) is adjusted so that the beam FEDI aims at theappropriate point on (4424).

Step 31. Fine Orientation Adjustment of the Two-sided Mirror (4424),Rotation and Translation Adjustment of (4427) and Adjustment of theFocal Length and Position of the Doublet (4425)(4426)

The position of (4425)(4426) must be adjusted so that the image of thebeam FEDI, when the control arrays$A_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

and $B_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

are applied respectively to the SLM (4405) and to the SLMs (4412) and(4417), is effectively a point image in the frequency plane in which(4427) is located. The position of (4424) and (4427) must be adjusted sothat, under these conditions, the lit point on the SLM (4427) has thecoordinates $\left( {\frac{N_{pix}}{2},\frac{N_{pix}}{2}} \right).$

The position of (4425)(4426)(4427) must moreover be adjusted so thatthere is a point-to-point correspondence between the pixels of (4417)and those of (4427), i.e. so that, when the control arrays A_(i,j) andB_(i,j) are applied respectively to the SLM (4405) and to the SLMs(4412) and (4417), the lit point on (4427) is the pixel of coordinates(i,j), whatever the integers i and j.

The frequency-meter is positioned between (4424) and (4428). Itspolarizer is put in place. The beam FEDI is used. The control array of(4427) is set to 0 and the polarizer of the frequency-meter is adjustedto cancel the intensity received on the CCD of the frequency-meter. Theseries of operations o21 to o23 is then repeated a sufficient number oftimes to converge towards a correct adjustment.

o21. Adjustment of Joint Position of (4425) and (4426) to Obtain a PointFrequency Image

The control array $A_{{\frac{N_{pix}}{2} - 1},\frac{N_{pix}}{2}}$

is applied to the SLM (4405) and the control array$B_{{\frac{N_{pix}}{2} - 1},\frac{N_{pix}}{2}}$

is applied to the SLMs (4412) and (4417). The control array applied to(4427) is set to 1. The position of the assembly (4425) (4426) isadjusted to have, on the CCD sensor of the frequency-meter, an image asclose as possible to a point image.

o22. Orientation Adjustment of (4424) and Translation Adjustment of(4427) to Obtain a Centered Frequency Image

(4424) and (4427) are adjusted so that the central frequency generatedis in the middle of the SLM. This adjustment is carried out with thefrequency-meter and with the maximum-location program PA defined above.However, in this maximum-location program, the SLM (4412) is replacedhere by the SLM (4427).

o23. Adjustment of (4425) to Obtain Proper Frequency Magnification andRotation Adjustment of (4427)

The program PC is used. However, in this program:

the SLM (4412) is replaced by the SLM (4427)

when a control word A_(i,j) is applied to the SLM (4405), thecorresponding control word B_(i,j) is applied in addition to the SLMs(4412) and (4417).

The inter-lens distance correction obtained is applied to the distancebetween (4425) and (4426) and the rotation adjustment is carried out on(4427).

Step 32. Two-axis Translation Adjustment of the Diaphragm (4428)

(4428) is mounted on a two-axis translation positioner allowingdisplacements in a plane orthogonal to the optical axis.

The aperture of (4428) is known. It must be adjusted in translation.

FEDI is used. (4428) is temporarily eliminated. An auxiliary CCD isplaced just behind (4428). The control array$A_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

is applied to the SLM (4405) and the control array$B_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

is applied to the SLMs (4412) and (4417). The control array of (4427) isat 1. The center of the illuminated area is marked on the auxiliary CCD.(4428) is then put in place so that its center coincides with the centerof the illuminated area.

Step 33. Rotation Adjustment of the Polarizer (4352)

This polarizer is mounted on a single-axis rotation positioner enablingthe adjustment of its rotation position around its optical axis.

FEDI is used. The control array of (4427) is set to 0. The controlarrays of the SLMs (4405)(4412)(4417) are not modified. The rotationposition of (4352) is adjusted to cancel the intensity received on theCCD (4339).

Step 34. Orientation Adjustment of the Mirror (4344)

This mirror is mounted on a positioner allowing its orientation to beadjusted. The diaphragm (4349) is open widely. A diffuser is used toverify that the beam FD reaches the CCD (4339) and is centered on thisCCD.

Step 35. Position Adjustment and Aperture of the Diaphragm (4349)

This diaphragm is mounted on a two-axis translation positioner allowinga displacement in the plane of the optical axis, and has an adjustableaperture.

Its position is adjusted in the presence of FRD so that the image of thediaphragm is centered on the CCD and its aperture is adjusted so thatthis image covers the entire CCD.

Step 36. Adjustment of the Focal Length and Position of the Doublet(4430)(4429) and Translation and Rotation Adjustment of the Sensor(4339)

The lenses (4430) (4429) are mounted on a translation positionerallowing, on the one hand, the joint displacement of the two lenses,and, on the other, the displacement of (4429) in relation to (4430).

The CCD (4339), integral with the camera (4384) is mounted on athree-axis rotation positioner allowing, on the one hand, rotationaround the optical axis and, on the other, adjustment of the orientationof the sensor, coupled with a three-axis translation positioner.

The positioners of the lenses (4430) (4429) and of the camera (4384) arethemselves mounted on a positioner allowing a displacement of theassembly in the direction of the optical axis.

The position of (4430)(4429) must be adjusted so that the image of thebeam FEDI, when the control arrays$A_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

and $B_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

are applied respectively to the SLM (4405) and to the SLMs (4412) and(4417), and when the control array of (4427) is set to 1, is effectivelya point image in the frequency in which (4339) is located. The positionof (4339) must be adjusted so that, under these conditions, the pointlit on the CCD (4339) has the coordinates$\left( {\frac{N_{pix}}{2},\frac{N_{pix}}{2}} \right).$

The position of (4339)(4429)(4430) must moreover be adjusted so thatthere is a point-to-point correspondence between the pixels of (4417)and those of (4339), i.e. so that, when the control arrays A_(i,j) andB_(i,j) are applied respectively to the SLM (4405) and to the SLMs(4412) and (4417), and when the control array of (4427) is at 1, thepoint lit on (4339) is the pixel of coordinates (i,j), whatever theintegers i and j.

The beams FEDI and FRD are used.

The array C applied to (4427) is set to 1.

The series of operations o31 to o33 is repeated a sufficient number oftimes.

o31. Adjustment of Joint Position of (4429) and (4430) to Obtain a PointFrequency Image

the control array $A_{{\frac{N_{pix}}{2} - 1},\frac{N_{pix}}{2}}$

is applied to the SLM (4405) and the control array$B_{{\frac{N_{pix}}{2} - 1},\frac{N_{pix}}{2}}$

is applied to the SLMs (4412) and (4417). The punctuality of the imagegenerated on the CCD (4339) is evaluated according to the proceduredescribed in 7.3.3.1. The position of the assembly (4429) (4430) isadjusted to have an image as close as possible to a point image.

o32. Translation Adjustment of (4339) to Obtain a Centered FrequencyImage

The translation position of (4339) is adjusted so that the coordinatesof the point image, calculated according to the procedure indicated in7.3.3.1., are $\left( {\frac{N_{pix}}{2},\frac{N_{pix}}{2}} \right).$

o33. Adjustment of (4429) to Obtain Correct Frequency Magnification, andRotation Adjustment of (4339)

The program PC is used. However, in this program:

when a control word A_(i,j) is applied to the SLM (4405), thecorresponding control word B_(i,j) is applied in addition to the SLMs(4412) and (4417).

the arrays X[i] and Y[i] are not obtained by the procedure PB. Theycorrespond to the coordinates of the maximum determined by the proceduredescribed in 7.3.3.1., without any specific action on an SLM.

The inter-lens distance correction obtained is applied to the distancebetween (4429) and (4430).

The rotation adjustment is carried out on (4339)(4384), rotating aroundthe optical axis.

The orientation of the CCD is adjusted to have$\frac{{X\lbrack 1\rbrack} - {X\lbrack 4\rbrack}}{{Y\lbrack 2\rbrack} - {Y\lbrack 3\rbrack}} = {\pm 1}$

Step 37. Adjustment of Translation Position of the Assembly Consistingof the CCD (4339) and the Doublet (4430)(4429)

The beam FRDI is used. An auxiliary CCD is put in place at the locationof the diaphragm (4313). The position of said assembly is adjusted sothat the image of FRDI on the auxiliary CCD is a point image.

Step 38. Orientation Adjustment of the Mirror (4347)

This mirror is mounted on a two-axis positioner enabling its orientationto be adjusted.

The control of the SLMs is unchanged. Between (4345) and (4346), twobeams propagating in opposite directions are superimposed. A diffuserintroduced on the side of the beam will be lit on both sides. The litparts of each side of the diffuser must be the same: the two beams arethen exactly superimposed. (4347) is adjusted so that, between (4345)and (4346), the beams propagating in both directions are exactlysuperimposed.

Step 39. Translation Adjustment of the Lens (4346)

The lens (4346) is mounted on a single-axis translation positionerallowing its position to be adjusted in the direction of the opticalaxis.

FRDI is used. The frequency-meter is positioned on the path of FRDIbetween (4345) and (4430). The position of the lens (4346) is adjustedso that the image on the CCD of the frequency-meter is a point image.

Step 40. Orientation Adjustment of the Semi-transparent Mirror (4345)

The semi-transparent mirror (4345) is mounted on a two-axis positionerallowing its orientation to be adjusted.

An auxiliary sensor is temporarily placed behind (4313). Thesemi-transparent mirror (4313) is adjusted so that the image on thistemporary sensor is centered in relation to the diaphragm (4313).

Step 41. Translation Position Adjustment of the Objectives (4317) and(4319)

The objective (4319) is mounted on a focussing device. The objective(4317) is mounted on a two-axis translation positioner allowingdisplacement in a plane orthogonal to the optical axis.

The beam FRDI is used. A temporary CCD sensor is positioned just behind(4323) on the path of FRDI. The position of the objectives is adjustedto obtain a centered point image.

Step 42. Introduction of a Temporary Plane Illuminating Beam

This beam, which will be called “FEP,” is derived from the sensorthrough a semi-transparent mirror placed between (4304) and (4305) andis brought by a set of mirrors to the entry of the microscope objective(4317), on the side of the sample. The objective (4319) must betemporarily eliminated for this purpose. At the input to the objective,this beam is directed along the optical axis of the objective.

Step 43. Orientation Adjustment of the Mirror (4314) and TranslationAdjustment of the Lens (4312)

The mirror (4314) is mounted on an angular positioner enabling itsorientation to be adjusted. The lens (4312) is mounted on a single-axistranslation positioner in the direction of the optical axis.

FEP and FRD are used. An image is obtained from the sensor (4339). Thepunctuality and the coordinates of the FEP image point are evaluated bythe procedure described in 7.3.3.1.

The lens (4312) is adjusted so that the image is a point image.

The mirror (4314) is adjusted so that the image is centered.

Step 44. Objective Position Adjustment

The beam FEP is eliminated and the objective (4317) is put in placeagain. The beam FRDI is used. A temporary CCD sensor is placed behind(4323) on the path of FRDI. The objectives are adjusted so that theimage is a point image and is centered in relation to the diaphragm(4323).

Step 45. Translation Adjustment of the Lens (4324)

FED is used. It passes successively through the objectives (4317) and(4319), then the lens (4324) and reaches the frequency-meter, which ispositioned behind (4324). The position of (4324) is adjusted to obtainan image as close as possible to a point image on the CCD of thefrequency-meter.

Step 46. Orientation Adjustment of the Mirrors (4432) (4435), FirstRotation Adjustment of (4436) and First Orientation Adjustment of theSemi-transparent Mirror (4325)

The SLMs (4405)(4412)(4417) are controlled so as to generate a centralfrequency. The beam FEG is used. (4325) is in the normal position. Thepath of the beam is controlled with a diffuser. Each mirror is adjustedso as to have the appropriate path. The beam must in particular occupythe entire aperture of (4323).

Step 47. Setup of Beam Extinguisher (4437)

This beam extinguisher is set up as indicated in 7.3.2.2.

Step 48. Adjustment of the Position and Focal Length of the Doublet(4433)(4434), Rotation Adjustment of the Assembly (4436), andOrientation Adjustment of the Semi-transparent Mirror (4325)

The lens (4434) is mounted on a translation positioner along the opticalaxis. This positioner and the lens (4433) are themselves mounted on asecond single-axis translation positioner along the optical axis.

The position of (4433)(4434) must be adjusted so that the image of thebeam FEG, when the control arrays$A_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

and $B_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

are applied respectively to the SLM (4405) and to the SLMs (4412) and(4417), and when the control array of (4427) is set to 1, is effectivelya point image in the frequency plane in which (4339) is located. Theposition of (4325) must be adjusted so that, under these conditions, theilluminated point on the CCD (4339) has the coordinates$\left( {\frac{N_{pix}}{2},\frac{N_{pix}}{2}} \right).$

The position of (4433)(4434)(4436)(4325) must moreover be adjusted sothat there is a point-to-point correspondence between the pixels of(4417) and those of (4339), i.e. so that, when the control arraysA_(i,j) and B_(i,j) are applied respectively to the SLM (4405) and tothe SLMs (4412) and (4417), and when the control array of (4427) is at1, the illuminated point on (4339) is the pixel of coordinates (i, j),whatever the integers i and j.

The beams FRRD and FEG are used. The control array applied to (4427) isset to 1.

This adjustment is carried out while performing a sufficient number oftimes the steps o41 to o43.

o41. Adjustment of the Joint Position of (4433) and (4434) to Obtain aPoint Frequency Image

The control array $A_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

is applied to the SLM (4405) and the control array$B_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

is applied to the SLMs (4412) and (4417). The punctuality of the imagegenerated on the CCD (4339) is evaluated according to the proceduredescribed in 7.3.3.1. The position of the assembly (4433) (4434) isadjusted to have an image as close as possible to a point image.

o42. Angular Adjustment of (4325) to Obtain a Centered Frequency Image

The angular position of (4325) is adjusted so that the coordinates ofthe point image, calculated according to the procedure indicated in7.3.3.1., are $\left( {\frac{N_{pix}}{2},\frac{N_{pix}}{2}} \right).$

o43. Adjustment of (4433) to Obtain the Proper Frequency Magnificationand Rotation Adjustment of (4436)

The program PC is used. However, in this program:

when a control word A_(i,j) is applied to the SLM (4405), thecorresponding control word B_(i,j) is applied to the SLMs (4412) and(4417).

the values X[i] and Y[i] are not obtained by the procedure PB. Theycorrespond to the coordinates of the maximum determined by the proceduredescribed in 7.3.3.1., without any specific action on an SLM.

The inter-lens distance correction obtained is applied to the distancebetween (4433) and (4434) and the rotation adjustment is carried out on(4436).

Step 49. Adjustment of the Rest of the “Left-hand” Part of theMicroscope

Each element still not adjusted corresponds to a symmetrical element inthe right-hand part of the microscope. The adjustment of the elementsnot yet adjusted is “symmetrical” with the adjustment of thecorresponding elements of the right-hand part of the microscope. It iscarried out symmetrically, the beam FEGI replacing the beam FEDI.However, the following must be taken into account:

a point centered at$\left( {\frac{N_{pix}}{2},\frac{N_{pix}}{2}} \right)$

 on the sensor (4339) can be obtained with control arrays$A_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

 and $B_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

 of the SLMs (4405),(4412),(4417) and with the beam FEG. It can also beobtained with the control arrays$A_{{\frac{N_{pix}}{2} - 1},\frac{N_{pix}}{2}}\quad B_{{\frac{N_{pix}}{2} - 1},\frac{N_{pix}}{2}}$

 and with the beam FEDI.

a point centered at$\left( {\frac{N_{pix}}{2},\frac{N_{pix}}{2}} \right)$

 on the sensor (4329) can be obtained with control arrays$A_{{\frac{N_{pix}}{2} - 1},\frac{N_{pix}}{2}}$

 and $B_{{\frac{N_{pix}}{2} - 1},\frac{N_{pix}}{2}}$

 of the SLMs (4405),(4412),(4417) and with the beam FED. It can also beobtained with the control arrays$A_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}\quad B_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

 and with the beam FEGI.

The control words of the SLMs are consequently not perfectly equivalentduring the two adjustments. The adjustment of the left part of themicroscope comprises in particular steps equivalent to the steps 31 and36. In these steps, the control arrays$A_{{\frac{N_{pix}}{2} - 1},\frac{N_{pix}}{2}}$

and $B_{{\frac{N_{pix}}{2} - 1},\frac{N_{pix}}{2}}$

must be replaced by $A_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

and $B_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}.$

Step 50. Setup of (4351)

This phase shift device is identical to the one described in 7.2.3. andis set up as indicated in 7.3.2.3. After this series of adjustments, thesystem is ready to be used.

8.6. Variant of utilization method

To generate the three-dimensional image of the object, we can confineourselves to the representation F_(0,0) defined in 7.17. This isequivalent, in the procedure described in 7.17.2., to adopting zeroarrays IB_(p,q) for any pair(p,q)≠(0,0).

It is also assumed here that that the object has an average index closeto the nominal index of the object observed and that the optical tableis totally free of vibrations.

The steps 7.9, 7.10, 7.11, 7.13, 7.15, 7.16 can then be eliminated. Thepresent method differs moreover from the preceding one in the methodused to adjust the position of the objectives before utilization, in thecontrol arrays applied to the beam deflector during imaging, and in theimage superposition algorithm.

8.6.1. Adjustment of objectives

This adjustment can be carried out in the presence of the object. It canalso be performed with a transparent plate provided the objective is notmoved when the object is introduced. If objectives designed to workwithout an immersion liquid and without a cover glass are used (nominalindex equal to 1) and if the sample is not very thick or has an averageindex close to 1, it may also be carried out in the absence of theobject. During this adjustment, the beams FEG and FRD are used and thefollowing operations are performed:

the control array of the SLM (4405) is set to 0

the control arrays of the other SLMs are set to 1

the rotation polarizer (4408) is adjusted so as to cancel the beam FEGreceived on (4339)

to the SLM (4405) is applied the control array$B_{\frac{N_{pix}}{2},\frac{N_{pix}}{2}}$

 which is zero at every point except at the point of coordinates$\left( {\frac{N_{pix}}{2},\frac{N_{pix}}{2}} \right).$

the punctuality of the image received on the sensor (4339) is evaluatedusing the procedure described in 7.3.3.1. with Fourier transformation

the position of the objectives is adjusted so that the image is aperfect point image and is centered

the polarizer (4408) is then returned to its initial position.

8.6.2. Control of beam deflector

The control array A_(ij) used in 8.4.1. for the SLM (4405) is replaced${{by}:{A_{ij}\left\lbrack {k,l} \right\rbrack}} = {E\quad \left( {{{\frac{1}{N_{pix}}\quad \left( {{k\quad \left( {i - \frac{N_{pix}}{2}} \right)} + {l\quad \left( {j - \frac{N_{pix}}{2}} \right)}} \right)}}{\% 2}} \right)}$

8.6.3. Algorithm for calculating the three-dimensional representation

Steps 1 and 2 of the algorithm described in 7.17.2. can be eliminated.In fact, the additional adjustment carried out, the control array of thebeam deflector used, and the absence of vibrations makes it possible toavoid any phase shift in the illuminating beam. In particular, if thevariant described in 7.18.5. is used, the phase shift device (4304) isnot used, the phase difference Θ_(d) being constant and can be chosen aszero.

9. FIFTH EMBODIMENT

This embodiment does not allow image acquisition as fast as in theprevious embodiment and is not, for this reason, the preferredembodiment in the general case. Nevertheless, in the particular case ofUV radiation, it constitutes the preferred embodiment. In fact, in thiscase, Embodiments 3 and 4 are not feasible owing to the non-availabilityof liquid crystals and polarizers. Since it can work with UV radiationof short wavelength, this embodiment is also the one which makes itpossible to obtain the best definition on the generated image.

9.1. Principles

This fifth embodiment is similar to the second embodiment in that theimage is captured in a spatial plane and in that the variations in thebeam direction are obtained by means of a mobile mirror. It is similarto the third embodiment in that two microscope objectives are used, andin that most of the algorithms are modified forms of those used in thethird embodiment. It differs from all the preceding embodiments in thatthe reference wave is not fixed but is modified at the same time as theilluminating wave. It is described in FIGS. 73 and 74.

The aim of this fifth embodiment is to improve resolution through theuse of an ultraviolet laser. No ferroelectric liquid crystals exist thatwork in the ultraviolet, so that consequently it is necessary to adoptsolutions with more conventional optical components. In particular, thebeam deflection device is a mobile mirror (5113).

However, the use of a mobile mirror in a system similar to the first orsecond embodiment generates vibrations. After each movement of themirror, it is necessary to wait for the system to stabilize beforeproceeding with the acquisition. To overcome vibrations caused by themobile mirror, it is necessary to position it outside the optical tableand to separate the illuminating and reference beams on the opticaltable after passing the mirror. Mirror movements thus result insimultaneous movement of the reference and illuminating beams.

In order to benefit fully from ultraviolet resolution possibilities, itmust be possible to carry out, as in the third and fourth embodiments,changes in the orientation of the electric field vector of theilluminating beam. These changes are made by separating the wave, bymeans of a semi-transparent mirror (5102), into two paths, a phase blade(5111) modifying the polarization being inserted on one of the paths andshutters (5104) and (5109) allowing the choice of the path used. The twowaves are then superimposed again by a mirror (5112). The shutters areplaced at a point where the light ray occupies only a small spatialextension and can thus be opened and closed rapidly.

For the same reason, it is necessary to have several analysisdirections. On each side of the microscope, two CCD sensors are used,one for each analysis direction. To the extent that it is difficult tohave good polarizers in the UV domain, the analysis direction will bemodified only by a modification in the reference wave polarizationdirection, for example by means of retardation plates (5238) (5239)which modify this polarization differently before each sensor.

The reference wave is mobile and can in particular pass through theobject at an angle close to the maximum aperture of the objective. Thespatial frequency thus received on the sensor can be twice as high aswith a system in which the reference wave is centered in relation to theoptical axis, as in the second embodiment. For the same image size, asensor with twice the dimensions in pixels is thus necessary, inrelation to the other embodiments,

To eliminate the direct illuminating wave, a glass (5165) or (5191) isused on which is fixed a black spot placed in a frequency plane. Bymoving this black spot, one modifies the eliminated frequency. On theother hand, this method does not give the value of the wave at the pointof direct impact of the illuminating wave, on which are based thethree-dimensional reconstructions used in the other embodiments.Moreover, the mobile mirror is poorly suited to quick changes in theilluminating wave between very different frequencies. And these changeswere necessary in the third embodiment for obtaining the referenceimages which allowed phase corrections for the two-dimensionalrepresentations obtained on the sensor opposite the point of directimpact of the illuminating wave. For this reason, a phase correctionmethod requiring neither a reference image nor acquisition of the pointcorresponding to the direct illumination must be provided for in thisembodiment.

Paragraph 9.2. describes materially the microscope used, and Paragraph9.3. gives the applicable dimensioning principles. As the microscope hasa material design differing significantly from Embodiment 3, itsadjustment and its utilization also differ considerably from theadjustment and utilization of the microscope according to Embodiment 3.

The microscope entails a series of adjustments carried out in theabsence of the sample:

Position adjustment for the different elements takes place as describedin Paragraph 9.5. This adjustment involves an image acquisitionprocedure described in Paragraph 9.4.

The arrays allowing the control of the beam deflection mirror aredetermined as described in Paragraph 9.6.

The arrays allowing the control of the glasses (5165) and (5191) used toeliminate the direct illuminating beam are determined as described inParagraph 9.7.

The constant K, equivalent to the one used in the first embodiment, isdetermined as described in 9.8.

The position of the CCD sensors entails a fine adjustment described inParagraph 9.10.

The array characterizing the frequency response of the sensors isdetermined as described in Paragraph 9.11.

The relative coordinates of the central points of the images obtained oneach side of the microscope are determined as described in Paragraph9.12.

The phases of each illuminating beam are determined as described inParagraph 9.13.

After placing the sample, the microscope undergoes a number ofadditional adjustments:

The position of objectives is adjusted as described in Paragraph 9.14.

The relative coordinates x,y,z of the origins of the reference beamsassociated with each objective, as well as the average index n_(o) ofthe sample and its thickness L, are determined as described in Paragraph9.15.

The value w₀ characterizing the position of the sample is calculated asdescribed in Paragraph 9.16. The procedure described in Paragraph 9.16.is essentially similar to the one described in Paragraph 7.15. andcomprises in particular a first focussing adjustment.

The aberration compensation function D_(p) is obtained as described inParagraph 9.17.

When these adjustments have been completed, the procedure for obtainingthree-dimensional images is started. A version of this procedure,similar to the one described for the third embodiment, is described in9.18. A specifically adapted version of this procedure is described in9.19. In every case, the two-dimensional frequency representations areacquired using a procedure described in Paragraph 9.9., which is alsoused in certain adjustment steps.

Although this is not mentioned, a focussing adjustment similar to theone described in 7.17.3. is carried out, which may involve arecalculation of w₀ and D_(p).

A very simplified version of the functioning of this microscope isdescribed in 9.20.

After algorithmic modifications similar to those described in 7.20 and7.21, this embodiment can be adapted to the use of objectives exhibitingaberrations.

9.2. Material description

An overall diagram of the system is represented in FIGS. 73,74,63. Theplane of FIGS. 73 and 74 is horizontal. The assembly (5176) surroundedby dotted lines in FIG. 74 is identical to the corresponding assembly inthe fourth embodiment and is represented in FIG. 63.

A laser polarized in the vertical direction (5100) produces a beam whoseelectric field vector is thus directed along an axis orthogonal to theplane of the figure. This beam then passes through a beam expander(5101).

The beam is then split in two by a semi-transparent mirror (5102). Oneof the beams coming from (5102) passes through a lens (5103) then ashutter (5104) placed in the focal plane of this lens. It then passesthrough a second lens (5105) whose front focal plane coincides with therear focal plane of (5103), and is then reflected by a mirror (5106) anda semi-transparent mirror (5112). The part of the beam which is notreflected by (5112) strikes an absorbing surface (5253). The second beamcoming from (5102) is reflected by a mirror (5107), passes through alens (5108) then a shutter (5109) placed in the focal plane of thislens. It then passes through a second lens (5110) whose front focalplane coincides with the rear focal plane of (5108). It then passesthrough a retardation plate (5111) and then through mirror (5112). Onexiting the mirror (5112) the two beams coming from the semi-transparentmirror (5102) are again superimposed. The focal lengths of the lenses(5103) (5105) (5108) (5110) are equal. The retardation plate (5111)introduces an optical path difference of a half-wavelength between itstwo neutral axes. It is positioned so as to transform the entering beampolarized in the vertical direction into a beam polarized in thehorizontal direction. The small spatial extension of the beam passingthrough the shutters (5104) (5109) allows the use of fast mechanicalshutters, a small displacement being sufficient to shutter the beam.

The beam coming from (5112) is directed towards the mirror (5113) whichreflects it. This mirror on a two-axis positioner (5114) similar to theone represented in FIGS. 2 and 3, which enables control of itsorientation. It is placed at the rear focal point of the lenses (5110)and (5105). The beam coming from (5113) then goes through the lens(5250) whose front focal point is on (5113), then the lens (5251) whosefront focal point coincides with the rear focal point of (5250). It isthen reflected by a partially transparent mirror (5115) which produces areference beam directed towards (5117). The semi-transparent mirror(5117) then splits the reference beam into a right reference beam FRDand a left reference beam FRG. The beam having passed through (5115)then goes through a partially transparent mirror (5116) which separatesa specific FS beam from it. The beam having passed through (5116) isthen split by a semi-transparent mirror (5118) into a right illuminatingbeam FED and a left illuminating beam FEG.

The beam FRD first passes through a lens (5120) whose front focal pointcoincides with the rear focal point of (5251), then a second lens (5121)whose front focal point coincides with the rear focal point of (5120).It is then reflected by a mirror (5122) placed at the rear focal pointof (5121) and mounted on a piezoelectric stack (5123) producingdisplacements of the order of the wavelength, which constitutes thephase shift device, on the same principle as the element (122) of FIG.1. It then passes through a filter (5124) allowing the adjustment of itsintensity, and then a lens (5125) whose front focal point is on themirror (5122). It then passes through a lens doublet (5127)(5126)operating according the principle explained in 8.1.4.1., the front focalpoint of the doublet (5127)(5126) coinciding with the rear focal pointof (5125). It is then split into two beams by a partially transparentmirror (5234). One of these beams is then reflected by the mirror(5235), passes through a retardation plate (5239), is then reflected inpart towards the CCD (5174) by the partially transparent mirror (5236),the non-reflected part being stopped by an absorbing surface (5237). Theother beam passes through a retardation plate (5238), is then reflectedin part towards the CCD (5171) by the partially transparent mirror(5232), the non-reflected part being stopped by an absorbing surface(5233). The CCDs (5174) and (5171) are mounted respectively on cameras(5175) and (5172). They are each in an rear focal plane of the doublet(5126)(5127).

The beam FRG is reflected by a mirror (5252). It passes through a lens(5145) whose front focal point coincides with the rear focal point of(5251), then a second lens (5146) whose front focal point coincides withthe rear focal point of (5145). It is reflected by a mirror (5147)placed at the real focal point of (5146) and mounted on a piezoelectricstack (5148) which constitutes the phase shift device. It then passesthrough a filter (5149) allowing the adjustment of its intensity, then alens (5150) whose front focal point is on the mirror (5147). It thenpasses through a lens doublet (5151)(5152) functioning according to theprinciple explained in 8.1.4.1., the front focal point of the doublet(5151)(5152) coinciding with the rearfocal point of (5150). It thenpasses through a rotation adjustment device of the type described in8.1.4.2., made up of the mirrors (5219) (5220) and of the assembly(5221) made up of the mirrors (5214) (5215) (5216) (5217), and mobile inrotation around an axis passing through the center of the mirrors(5220)(5214)(5217). It is then split into two beams by a partiallytransparent mirror (5244). One of these beams is then reflected by themirror (5245), passes through a retardation plate (5249), is thenreflected in part towards the CCD (5198) by the partially transparentmirror (5246), the non-reflected part being stopped by an absorbingsurface (5247). The other beam passes through a retardation plate(5248), is then reflected in part towards the CCD (5201) by thepartially transparent mirror (5242), the non-reflected part beingstopped by an absorbing surface (5243). The CCDs (5198) and (5201) aremounted respectively on the cameras (5199) and (5202). They are each inan rear focal plane of the doublet (5151)(5152).

The beam FED is reflected by a mirror (5141), passes through a filter(5142) allowing the adjustment of its intensity, a lens (5143) whosefront focal point coincides with the rear focal point of (5251), ashutter (5144), and a lens (5154) whose front focal plane coincides withthe rear focal plane of (5143) and whose rear focal plane coincides withthe image of the diaphragm (5158) through the lens (5157). It isreflected by mirrors (5153) and (5155), then split into two beams by apartially transparent mirror (5156). One of the beams, directed towards(5157), is the main beam and will be denoted FED. The other beam,directed towards (5159), constitutes the opposite indicator of FED andwill be denoted FEDI.

The beam FED passes through the lens (5157), the diaphragm (5158), thedevice (5176) represented in FIG. 63, passes through the diaphragm(5184) and the lens (5183). A focal plane of the lens (5157) coincideswith the rear focal plane of the microscope objective (4317) [in theconventional direction of the use of the objective, here opposite to thedirection of the rays]. A focal plane of the lens (5183) coincides withthe rear focal plane of the microscope objective (4319). The diaphragms(5158) and (5184) are placed in the planes in which the objectivesnormally form the images of the sample. The beam coming from (5183)passes through the semi-transparent mirror (5182). It passes through thelens (5188) whose front focal plane coincides with the image of thediaphragm (5184) through the lens (5183). It then passes through a glass(5191) of small thickness, on which there is an absorbing black spot afew tens of micrometers in diameter. This glass is positioned in therear focal plane of the lens (5188) and is designed to stop the directilluminating beam. A diaphragm (5190) placed roughly in the same planeimproves the filtering of spatial frequencies carried out by themicroscope objective. The beam then passes through a doublet(5192)(5193) whose front focal plane coincides with the glass (5191). Itis then split into two beams by a semi-transparent mirror (5240). One ofthe beams from (5240) reaches the CCD (5198) after having passed throughthe partially transparent mirror (5246). The other beam is reflected bythe mirror (5241) and reaches the CCD (5201) after having passed throughthe partially transparent mirror (5242). The CCDs (5198) and (5201) areeach in an rear focal plane of the doublet (5192)(5193).

The beam FEDI passes through the lens (5159) whose focal plane coincideswith the rear focal plane of the lens (5154). It reaches the mirror(5160), optionally closed off by the shutter (5161), which reflects it.It again passes through the lens (5159) and is reflected by thesemi-transparent mirror (5156) towards the lens (5162). It passesthrough the lens (5162) whose front focal plane coincides with the imageof the diaphragm (5158) through the lens (5157). It then passes throughthe partially transparent mirror (5163). Then it passes through a glass(5165) of small thickness on which there is an absorbing black spot afew tens of micrometers in diameter. This glass is positioned in therear focal plane of the lens (5162) and is designed to stop the directilluminating beam. An optional diaphragm (5164) placed roughly in thesame plane improves the filtering of spatial frequencies carried out bythe microscope objective. The beam then passes through a lens (5166)whose front focal plane coincides with the rear focal plane of (5162).It is then split into two beams by a semi-transparent mirror (5230). Oneof the beams coming from (5230) reaches the CCD (5174) after havingpassed through the partially transparent mirror (5236). The other beamis reflected by the mirror (5231) and reaches the CCD (5171) afterhaving passed through the partially transparent mirror (5232). The CCDs(5174) and (5171) are each in an rear focal plane of the lens (5166).

The beam FEG is reflected by the mirrors (5119) (5204), passes through afilter (5205) enabling its intensity to be adjusted, a lens (5206) whosefront focal point coincides with the rear focal point of (5251), ashutter (5207), a doublet (5179)(5178) whose front focal plane coincideswith the rear focal plane of (5206) and whose rear focal plane coincideswith the image of the diaphragm (5184) through the lens (5183). It thenpasses through a rotation adjustment device of the type described in8.1.4.2., made up of mirrors (5177) (5180) and of the assembly (5181)made up of the mirrors (5210) (5211) (5212) (5213), and mobile inrotation around an axis passing through the center of the mirrors (5180)(5210) (5213). It is then split into two beams by a partiallytransparent mirror (5182). One of the beams, directed towards (5183), isthe main beam and will be denoted FEG. The other beam, directed towards(5185), constitutes the reverse indicator of FFG and will be denotedFFG1.

The beam FEGI passes through the lens (5185) whose focal plane coincideswith the rear focal plane of the doublet (5178) (5179). It reaches themirror (5187), optionally closed off by the shutter (5186), whichreflects it. It then passes again through the lens (5185) and isreflected by the semi-transparent mirror (5182) towards the lens (5188).

The beam FS first passes through a filter (5128), then a lens (5129) andis reflected by a mirror (5130). It then passes through a shutter(5254), then a lens (5140) whose front focal point coincides with therear focal point of (5129). It is then split by a semi-transparentmirror (5163) into a main beam FS directed towards (5162) and anopposite indicator beam directed towards (5189), which will be denotedFSI. The beam FSI passes through a lens (5189), is reflected on a mirrorplaced at the front focal point of (5189) and capable of being closedoff by a shutter (5209), passes again through the lens (5189) and isagain reflected by the semi-transparent mirror (5163) in the directionof (5164).

Each lens used is an achromat or a composite lens minimizing opticalaberrations.

The beam control principles indicated in 8.1.1. and 8.1.4. remain valid,and the spatial (E) and frequency (F) planes have been indicated in thesame manner as for the fourth embodiment. The reference wave here isparallel, as is the illuminating wave and as in the second embodiment.For this reason it is a point wave in the frequency planes and parallelin the spatial planes like the illuminating wave. The special beam FSis, by contrast, a point beam in the spatial planes and parallel in thefrequency planes, as was the reference wave in the fourth embodiment.

Many of the elements are mounted on positioners enabling the adjustmentof their position in an adjustment phase.

The assemblies (5181) and (5221) are mounted on rotation positioners, inaccording with their operating mode explained in 8.1.4.2.

The other mirrors, partially transparent and piezoelectric mirrors, aremounted on angular positioners allowing the adjustment of theirorientation.

The lenses which will be adjusted in 9.5. are mounted onsingle-dimension positioners allowing a displacement in the direction ofthe optical axis.

Each doublet is made up of two lenses. It transforms a frequency planelocated on one side of the doublet into a spatial plane located on theother side of the doublet. The lens located on the side of the spatialplane is mounted on a positioner allowing translation in the directionof the optical axis. This positioner is itself mounted on a secondpositioner also allowing translation in the direction of the opticalaxis. The second lens is mounted directly on this second positioner.

The objective (4317) is mounted on a two-dimension positioner allowingpositioning in a plane orthogonal to the optical axis. The objective(4319) is mounted on a focussing device. The sample (4318) is mounted athree-axis translation positioner.

The CCDs are mounted on single-axis rotation positioners and three-axistranslation positioners allowing rotation around the optical axis andthree degrees of freedom in translation.

The broken line (5203) separates two zones. The elements located on theleft of this line are mounted on a table linked directly to the groundwithout shock absorption. The elements located on the right of this lineor in FIG. 74 are mounted on an optical table suitably isolated fromvibrations. The two arrays are at the same level. The glasses (5165) and(5191), the diaphragms (5164) and (5190), and the shutters (5144) (5207)are the only exceptions to this rule. Each of the glasses (5165) and(5191) is mounted on a powered two-axis translation positioner allowingdisplacement in a plane orthogonal to the optical axis, itself mountedon a manual positioner manual allowing translation in the direction ofthe optical axis, itself linked directly to the ground. The two-axistranslation positioning system must be precise and not produce spuriousrotational displacements of the glass. Such displacements would lead tophase variations that may in certain cases be detrimental to the qualityof the images produced.

Each of the shutters (5144) (5207) is linked directly to the ground.Each of the diaphragms (5164) (5190) is linked to the floor via athree-axis translation positioner and a single-axis rotation positioneraround the optical axis.

In order to be able to link the glasses and shutters to the floor, arigid mechanical construction is used to obtain a stable support locatedover the optical table, linked to the ground and not to the opticaltable, and on which can be fixed the shutters as well as the glasses viatheir positioners.

The diaphragms (5164) and (5190) are made up as indicated in FIG. 82.The diaphragm (5710) has a circular aperture (5711), a part (5712)allowing the shuttering of an additional surface portion in thisaperture.

On each camera has been indicated the system of coordinates used toexpress the coordinates of the pixels of the corresponding CCD. On eachretardation plate has been indicated a system of coordinates. Thedirection vector of this system in the plane of the figure is denoted{right arrow over (i)} and the direction vector of this system in theplane orthogonal to the figure is denoted {right arrow over (j)}. Theretardation plate (5111) has a neutral axis directed along {right arrowover (i)}+{right arrow over (j)}. The other retardation plates haveneutral axes directed along {right arrow over (i)} cos π/8+{right arrowover (j)} sin π/8.

If this microscope operates in the ultraviolet domain:

All the non-polarizing components traversed by the light, i.e. thelenses, including those of the objectives, the glasses, and thesubstrates of semi-transparent mirrors, can be made of a silica.Objectives in silica or in quartz are available from variousmanufacturers.

The mirrors and semi-transparent mirrors must be designed specially forUV.

The retardation plate are for example in quartz.

The laser is for example an excimer laser. In pulsed mode, the pulsesmust be synchronized with image acquisition, for example one image perpulse.

9.3. Dimensioning

The following notation is used:

f₁: focal length of the lens (5162) or of the lens (5188)

f₂: focal length of the lens (5166) or of the doublet (5192,5193)

f₃: focal length of the lens (5157) or of the lens (5183)

p_(c) distance between centers of two adjacent pixels on the CCD sensors

2 N_(pix): side dimension in pixels of a CCD sensor

o: numerical aperture of a microscope objective

g: magnification of a microscope objective

f_(o): focal length of a microscope objective.

d_(o): distance between the lens (5157) and the diaphragm (5158).

l₁: width of the diaphragm (5164) or (5190)

The lens (5157) must have its front focal plane coincident with the rearfocal plane of the microscope objective, and is located at a distanced_(o) from the image of the objective. We check that this means:

f ₃ =gf _(o) +d _(o)

The aperture of the beam upon reaching the CCD is:$\alpha = {\frac{f_{1}}{f_{2}}\quad \frac{o}{g}}$

The sampling period required on the CCD must be greater than p_(c)and isequal, by applying the Nyquist criterion, to:$\frac{\lambda}{2}\quad \frac{1}{2\quad \alpha}$

We thus obtain:$p_{c} \leq {\frac{\lambda}{4}\quad \frac{f_{2}}{f_{1}}\quad \frac{g}{o}}$

where$\frac{f_{1}}{f_{2}} \leq {\frac{\lambda}{4p_{c}}\quad \frac{g}{o}}$

The ratio f₁/f₂ must thus be equal or slightly lower than${\frac{\lambda}{4p_{c}}\quad \frac{g}{o}},$

each of the values f₁ and f₂ being moreover sufficient to preventspherical aberration.

The width l₁ of the diaphragm (5164) must filter only the frequencieshigher than the nominal aperture of the objective. We must thus havel₁=2f₁o/g.

The diameter of the beam incident on the mirror (5113) must be such thatthe required direction variations are in fact obtainable in areproducible manner by means of the positioning system of this mirror.We denote as p_(α) the angular displacement interval of the mirror, i.e.the smallest variation of the angle or orientation of this mirrorobtainable in a reproducible manner by the positioning system of thismirror. We denote as D_(diaph) the aperture diameter of the diaphragm(5158) and as D_(mir) the diameter of the beam incident on the mirror(5113). We then check that the present condition is expressed by:$p_{\alpha}{\operatorname{<<}\frac{o}{{gN}_{pix}}}\quad \frac{D_{diaph}}{D_{mir}}$

in which the sign <<means << much smaller >>, or for example$p_{\alpha} < {\frac{1}{10}\quad \frac{o}{{gN}_{pix}}\quad {\frac{D_{diaph}}{D_{mir}}.}}$

In addition to the preceding criteria, a fairly large amount of libertyexists for the choice of the focal lengths of the other lenses. Thedimensioning criteria applied to the path of the beams FEG,FED,FRG,FRDare the following:

(1). The succession of the spatial and frequency planes must be asindicated in the diagram. This succession of spatial and frequencyplanes constitutes the method used to control the path and aperture ofthe beam as indicated in 8.1.1.

(2). The illuminating and reference beams, in the absence of thediaphragm used between the mirror (5113) and the CCD sensors, must havethe same diameter when they reach the sensors. Equivalently, they musthave the same aperture, the aperture here being the angle between theparallel beams reaching the CCDs for two different positions of themirror (5113).

(3). The focal lengths of the lenses must be sufficient to preventspherical aberration.

(4). The focal lengths of the different lenses are adapted in order tocomply with dimensional constraints.

For example, if the diameter of the beam is D₁ at the level of (5122)and must be D₂ at its arrival on the CCD sensor, and if the distancebetween (5122) and (5174), following the path adopted for the beam, isL, then if f_(a) and f_(b) are respectively the focal lengths of thelens (5125) and of the doublet (5126)(5127), and given the principlesset forth in 8.1.1., the criterion (1) is$\frac{D_{2}}{D_{1}} = \frac{f_{b}}{f_{a}}$

and the criterion (4) is 2f_(a)+2f_(b)=L, yielding$f_{a} = {\frac{L}{2}\quad \frac{D_{1}}{D_{1} + D_{2}}}$

and $f_{b} = {\frac{L}{2}\quad {\frac{D_{1}}{D_{1} + D_{2}}.}}$

The entire system will be adjusted so that, when the beam FEG penetratesinto the objective (4317), being directed along the optical axis, itsdirection upon arrival on (5174) or (5171) coincides with that of thereference wave. When the mirror (5113) is to be displaced from thiscentral position, the direction of the reference beam will be modifiedin one direction and the direction of the illuminating beam will bemodified in the opposite direction, i.e. in a frequency plane the pointscorresponding to the beams FEG and FED will remain symmetrical inrelation to the point corresponding to the illuminating beam when itenters the objective in the direction of the optical axis. Themaintenance of this symmetry is made possible by:

the general configuration of the apparatus. In fact, following thediagram in FIG. 67, when the beam passes from one spatial plane (4801)to a second spatial plane (4804), its direction is reversed. In theadopted configuration, the difference between the number of spatialplanes traversed by the illuminating beam and the number of spatialplanes traversed by the reference beam is an odd number, and theilluminating beam is thus reversed in relation to the reference beam.

compliance with condition (3), which means that the displacement of thereference beam and the displacement of the illuminating beam are of thesame amplitude.

This displacement symmetry allows the simplification of the algorithmsand the adjustment procedure.

The dimensioning criteria applied to the path of the beam FS are thefollowing:

the beam FS must be parallel upon its arrival on (5163)

its width upon its arrival on (5163) must be equal to that of a beamwhich would be a point beam in the plane (5158) and whose aperture wouldbe limited by the aperture of the objective. This width, upon thearrival of the beam on (5163), is equal to about f₁o/g.

9.4. Obtaining a simple two-dimensional frequency representation anddetermining the maximum

What is meant here by a simple two-dimensional frequency representationis a representation for which no account has been taken ofpolarizations, and which can be obtained without knowing the values of Kand of the point of direct impact of the illuminating wave, from asingle sensor. The steps in producing such a representation, for a givenilluminating wave, are the following:

Step 1—Acquisition

(5104) is open and (5109) is closed, so that the polarization of theilluminating wave is fixed.

d is defined as follows, the phase shift being produced with thepiezoelectric actuators (5123) (5148) previously calibrated:

Index d Phase shift(degrees) 0 +120 1 0 2 −120

The image is obtained from any one of the CCD sensors. When it isobtained by means of (5171) or (5201), the phase retardation plates(5238) and (5248) must be eliminated.

One thus obtains the arrays MF[d][i,j] in which i and j vary from 0 to2N_(pix)−1.

Step 2—Calculation of Two-dimensional Spatial Representations

The program performs:

MG[i, j]=1/6(2MF[0][i, j]−MF[1][i, j]−MF[2][i, j])+{tilde over(j)}1/2{square root over (3)}(MF[1][i, j]−MF[2][i, j])

Step 3—Fourier Transformation

The Fourier transform of the array MG according to the indices i andj isperformed. This generates the frequency representation MH[i, j] in whichp takes on the values of 0 or 1 and in which i andj vary from 0 to2N_(pix)−1. In certain cases, this Fourier transformation may beomitted: one then obtains a spatial representation instead of afrequency representation.

When the image is roughly a point image and the coordinates and thevalue of the maximum must be known, the maximum calculation programproceeds as indicated in 7.3.3.1., except that the array S now has thedimensions 2N_(pix)×2N_(pix) and is the array MH calculated as above. Itthus obtains the coordinates imax,jmax of the maximum on the sensorconcerned. The image is considered to be centered if (imax,jmax)=(N_(pix), N_(pix)) and it is considered to be a perfect pointimage when the value of the maximum obtained is the highest possiblevalue.

9.5. Adjustment of manual Positioners

9.5.1. Adjustment criteria

FEP designates a plane illuminating beam entering the objective (4317),being directed along the optical axis and reaching the sensors (5174)(5171).

The adjustments are designed to ensure that:

(1) the beams follow the appropriate path.

(2) the illuminating and reference beams are point beams in thefrequency planes and parallel in the spatial planes.

(3) the beams FS and FSI are point beams in the spatial planes

(4) the beams FS and FSI have point images centered on the sensors(5171) (5174) (5201) (5198).

(5) a parallel beam FEP entering a microscope objective (4317) anddirected along the optical axis has a point image centered on thetwo-dimensional frequency representations obtained using the procedure9.4. from images received on the sensors (5174) or (5171).

(6) When the image of the beam FEG on the two-dimensional frequencyrepresentations obtained using the procedure 9.4. from images receivedon sensors (5174) or (5171) is a point image and is centered, then theimage of the beam FEGI on the two-dimensional frequency representationsobtained using the procedure 9.4. from the images received on thesensors (5198) or (5201) is a point image and is centered.

(7) Whatever the position of the mirror (5113):

the points corresponding to the beams FRD, FEDI, FEG, FEP on thetwo-dimensional frequency representations obtained with the procedure9.4. from the images received on the sensors (5174) or (5171) arearranged as indicated in FIG. 78

the points corresponding to the beams FRG, FEGI, FED on thetwo-dimensional frequency representations obtained with the procedure9.4. from the images received on the sensors (5198) or (5201) arearranged as indicated in FIG. 79.

To explain further this condition, we shall denote in the same manner abeam and the corresponding point on one of the sensors and we shalldenote as (A,B) the vector linking the points A and B. This conditionmeans that:

(i) FRD and FEDI are on the same vertical line.

(ii) FEDI and FEG are on the same horizontal line.

(iii) FEP is in the middle of FRD and FEG

(iv) (FRG,FEGI)=(FRD,FEDI)

(v) (FEGI,FED)=(FEG,FEDI)

The adjustments result naturally from the compliance with conditions (1)to (7). The rest of the adjustment steps given in detail belowconstitute an example of the ordering of these adjustments.

9.5.2. Adjustment steps

In certain adjustment phases, a temporarily parallel beam denoted FEPwill be used. This beam is derived directly from the (5100) by means ofa semi-transparent mirror and directed towards the input of theobjective (4317), which it reaches while being directed along theoptical axis, and which it traverses before going to the sensors (5171)and (5174).

The setup of FEP requires the temporary elimination of the objective(4319). The parts of the system said to be “linked to the ground” are infact linked to a flat support ordinarily placed on the ground or on atable without any particular precautions, the optical table itself beingplaced, via shock absorbers, on this flat support. During all theadjustments described in the present paragraph, the optical table willbe connected to the flat support, i.e. fixed without shock absorptionand without freedom of movement to the flat support, in a position asclose as possible to the position of the optical table when it is “free”on its shock absorbers. The flat support itself will be fixed on asecond optical table. This device makes it possible to generateinterference patterns using the beam FEP, which would be impossiblebecause of the vibrations if said flat support were fixed directly tothe ground as in the normal operating phase of the microscope.

The use of the beams FEG,FED,FEGI,FEDI,FS,FSI is controlled by theshutters (5144) (5207) (5218) (5209) (5161) (5186) (5254). Shutters notrepresented also allow the elimination of the beams FRD and IFRG. On animage of dimensions 2N_(pix)×2N_(pix) a point will be said to becentered if its coordinates are (N_(pix), N_(pix)).

The following steps constitute an example of the ordering ofadjustments:

Step 1: Pre-adjustment

A pre-adjustment is carried out, during which the mirror is positionedin a central position and the path of the beam is controlled with adiffuser. During this pre-adjustment, the position of all the elementsis adjusted so that the beam follows approximately the appropriate path.For example, at the exit of (5112), the proper spatial superposition ofthe beams is checked. Between (5156) and (5159), we check that thereflected beam ahs the same spatial extension as the arriving beam.

During all the adjustments which follow, this pre-adjustment may beconstantly verified or refined, with the aid of the diffuser, withoutthis being restated. In the case of work in the UV domain, it isdangerous to display the light directly. The diffuser is then replacedby an auxiliary CCD, and the area illuminated on this CCD is observed ona screen instead of observing it directly.

Step 2: Translation Adjustment of the Lens (5105)

The frequency-meter is positioned behind the lens (5105) and theposition of the lens is adjusted to have a point image on thefrequency-meter.

Step 3: Translation Adjustment of the Lens (5110)

The frequency-meter is positioned behind the lens (5110) and theposition of the lens is adjusted to have a point image on thefrequency-meter.

Step 4: Orientation Adjustment of the Semi-transparent Mirror (5112)

The frequency-meter is used and positioned behind (5112). The shutters(5104) and (5106) being opened and closed alternately, the integratedimage produced on the sensor of the frequency-meter consists of twopoints coming from each of the superimposed beam. The angular positionof (5112) is adjusted so as to superimpose these two points.

Step 5: Translation Adjustment of the Lens (5251)

The frequency-meter is positioned behind the lens (5251) and theposition of the lens is adjusted to have a point image on thefrequency-meter.

Step 6: Translation Adjustment of the Lens (5121)

The frequency-meter is positioned behind the lens (5251) and theposition of the lens is adjusted to have a point image on thefrequency-meter.

Step 7—Translation Adjustment of the Doublet (5127) (5126)

The frequency-meter is placed behind (5127) on the path of the beam andthe doublet is moved so that the image on the CCD of frequency-meter isa point image.

Step 8—Translation Adjustment of the Lens (5146)

This adjustment is carried out by means of the frequency-meter, placedbehind (5146). The image on the CCD of the frequency-meter must be asclose as possible to a point image.

Step 9—Translation Adjustment of the Doublet (5151) (5152)

The frequency-meter is placed behind (5152) on the path of the beam andthe adjustment is carried out so that the image on the CCD of thefrequency-meter is a point image.

Step 10—Translation Adjustment of the Lens (5140)

The adjustment is carried out by means of the frequency-meter placedbehind (5140). The image on the CCD of the frequency-meter must be apoint image.

Step 11—Translation Adjustment of the Lens (5154)

The frequency-meter is placed behind (5154). The image on the CCD of thefrequency-meter must be a point image.

Step 12—Translation Adjustment of the Doublet (5178) (5179)

The frequency-meter is placed for example behind (5213). The image onthe CCD of the frequency-meter must be a point image.

Step 13—Translation Adjustment of the Lens (5159)

The frequency-meter is placed between (5156) and (5162). The position of(5159) is adjusted so that the image of FEDI on the CCD of thefrequency-meter is a point image.

Step 14—Introduction of a Temporary Parallel Illuminating Beam FEP

This beam has the characteristics described at the beginning of thepresent paragraph.

Step 15—Translation Adjustment of the Lens (5157)

The frequency-meter is placed between (5156) and (5162). The position of(5157) is adjusted so that the image of FEP on the CCD sensor of thefrequency-meter is a point image.

Step 16—Orientation Adjustment of the Semitransparent Mirror (5156)

The position of (5156) is adjusted so that FEP and FEDI have pointimages coinciding on the frequency-meter.

Step 17—Translation Adjustment of the Lens (5162)

A CCD is temporarily placed at the location of (5158). FS is used.(5162) is adjusted so that the image of FS on this temporary CCD is apoint image.

Step 18: Orientation Adjustment of the Partially Transparent Mirror(5163)

The CCD remains used with the beam FS. (5163) is adjusted so that theimage of FS on this temporary CCD is centered in relation to thelocation of the diaphragm.

Step 19—Translation Adjustment of the Lens (5189)

The frequency-meter is positioned between (5163) and (5164). (5189) isadjusted to obtain a point image of FSI on the frequency-meter.

Step 20—Translation Adjustment of the Lens (5166)

FEDI is used. The frequency-meter is positioned on the path of the beambehind (5166) and the position of (5166) is adjusted so that the imageof FEDI on the frequency-meter is a point image.

Step 21—Three-axis Translation Adjustment of the Sensors (5174) and(5171)

FSI and FRD are used. A program calculates the two-dimensional frequencyrepresentation received on each CCD (5174) or (5171) according to theprocedure described in 9.4., but without carrying out Step 3 with itsFourier transformation. The position of the CCDs is adjusted to obtain,from each CCD, a centered point image.

Step 22—Orientation Adjustment of the Partially Transparent Mirror(5236)

FRD and FEDI are used. A program calculates the two-dimensionalfrequency representation received on the CCD (5174) according to theprocedure described in 9.4.

The position of (5236) is adjusted so that the two-dimensional frequencyrepresentation obtained on each CCD is centered.

Step 23—Orientation Adjustment of the Partially Transparent Mirror(5232)

FRD and FEDI are used. A program calculates the two-dimensionalfrequency representation received on the CCD (5171) according to theprocedure described in 9.4.

The position of (5232) is adjusted so that the two-dimensional frequencyrepresentation obtained on each CCD is centered.

Step 24—Adjustment of the Focal Length of the Doublet (5126) (5127) andRotation Adjustment Around the Optical Axis of the Sensors (5174) (5171)

The beams FRD, FEP are used. A program calculates the two-dimensionalfrequency representation received on each CCD (5174) o′ (5171) accordingto the procedure described in 9.4.

The initial position of the mirror (5113), in which the precedingadjustments were carried out, is the centered position. It must berecorded. It will be called position C.

The mirror (5113) is moved so that, on the two-dimensional frequencyrepresentation obtained on each CCD (5174) or (5171), the pointcorresponding to FEP is off centered as much as possible. This positionof the mirror is recorded and will be used again later. It will becalled position E.

The beam FEDI is introduced.

The positions of the different elements are adjusted so that, on each ofthe two representations obtained:

the coordinates of the points associated with FRID (central point) andFEDI are correct in relation to those of the point associated with FEP,i.e. symmetrical in relation to the horizontal axis passing through thepoint associated with FEP, as indicated in FIG. 78 where (5501)represents the contour of the two-dimensional area obtained after theprocedure described in 9.4. and in which (5502) represents the limitdefined by the aperture of the objective.

the images of FEP and FEDI remain point images.

 More precisely:

the focal length of the assembly (5126) (5127) is adjusted so that thedistance between the point associated with FEP and the point associatedwith FRRD is equal to the distance between the point associated with FEPand the point associated with FEDI. As the changes in the position ofthe lenses of the doublet can produce a loss of focus, the positions ofthe lenses of the doublet are also adjusted so that the image of FEP andFEDI remains a point image.

the rotation position of the sensors is adjusted so that the linepassing through the points associated respectively with FRD and FEDI isvertical.

Owing to the non-coincidence of the axis of rotation of the CCDs withthe central point of these sensors, this rotation adjustment can resultin a loss of the translation adjustment of the CCDs. For this reason,after this operation, the mirror is brought back to position C and Step21 is carried out again. Steps 21 and 24 can thus be repeated insequence a certain number of times so as to converge towards a correcttranslation and rotation adjustment of the sensors.

Step 25—Translation Adjustment of the Position of Objectives (4319) and(4317)

The mirror (5113) is brought back to its original position (position C),the beam FEP is eliminated, the objectives are put back in place. Atemporary CCD is positioned at the location of (5184). The beam FS isthe only one used. A transparent plate is used as the object.

The position of the objectives is adjusted to obtain a centered pointimage.

Step 26—Translation Adjustment of the Lens (5183)

The beam FED is used. The frequency-meter is positioned behind (5183) onthe path of FED. (5183) is adjusted so that FED has a point image on thefrequency-meter.

Step 27—Translation Adjustment of the Doublet (5178) (5179) andOrientation Adjustment of the Semi-transparent Mirror (5182)

The beams FEG and FRD are used. A two-dimensional frequencyrepresentation is obtained from the CCDs (5174) or (5171) by theprocedure 9.4.

The joint position of the doublet (5178) (5179) is adjusted so that therepresentation obtained is a point representation. The semi-transparentmirror (5182) is adjusted so that this image is centered.

Step 28—Adjustment of Focal Length of Doublet (5178)(5179), RotationAdjustment of the Assembly (5181), and Orientation Adjustment of theSemi-transparent Mirror (5182).

The beams FRD, FEG are used. A two-dimensional frequency representationis obtained from the CCDs (5174) or (5171) by the procedure 9.4. Theoperations o1 and o2 above must be repeated in this order a certainnumber of times to converge towards a correct adjustment.

o1: The Mirror (5113) is Put in Position C and the SemitransparentMirror (5182) is Adjusted so that the Point of the FrequencyRepresentation Corresponding to FEG is Centered

o2: The Mirror (5113) is Put in Position E (Offcentered)

The different elements are adjusted so that the point corresponding toFEG obtained on the image is correctly positioned in relation to theothers. The point corresponding to FEP is known by the adjustmentcarried out in Step 24. The point corresponding to FEG must besymmetrical with the point corresponding to FRD in relation to the pointcorresponding to FEP as indicated in FIG. 78.

More precisely:

the focal length of the doublet (5178)(5179) is adjusted so that thedistance between the points corresponding to FRD and FEP is equal to thedistance between the points corresponding to FEP and FEG. As changes inthe position of the doublet lenses can produce a loss of focus, thepositions of the doublet lenses are also adjusted so that the image ofFEG remains a point image.

the rotation position of the assembly (5181) is adjusted so that thepoints FRD, FEP and FEG are aligned.

Step 29—Translation Adjustment of the Lens (5185)

The mirror (5113) is brought back to position C. The frequency-meter ispositioned between (5182) and (5188). The position of (5185) is adjustedso that the image of FEGI on the frequency-meter is a point image.

Step 30—Translation Adjustment of the Lens (5188)

The beam FS is used. The frequency-meter is positioned behind (5188).The position of (5188) is adjusted so that FS has a point image on theCCD of the frequency-meter.

Step 31—Translation Adjustment of the Doublet (5192)(5193)

FEGI is used. The frequency-meter is positioned behind (5193). The jointposition of (5192)(5193) is adjusted so that FEGI has a point image onthe CCD of the frequency-meter.

Step 32—Three-axis Translation Adjustment of the Sensors (5198) and(5201)

FS and FRG are used. A program calculates the two-dimensional frequencyrepresentation received on each CCD (5198) or (5201) according to theprocedure 9.4., but without carrying out Step 3 with its Fouriertransformation. The position of the CCDs is adjusted to obtain from eachCCD a centered point image.

Step 33—Orientation Adjustment of the Partially Transparent Mirror(5246)

FRG and FEGI are used. A program calculates the two-dimensionalfrequency representation received on the CCD (5198) according to theprocedure 9.3.1.

The position of (5246) is adjusted so that the two-dimensional frequencyrepresentation obtained is centered.

Step 34—Orientation Adjustment of the Partially Transparent Mirror(5242)

FRG and FEGI are used. A program calculates two-dimensional frequencyrepresentation received on the CCD (520 1) according to the procedure9.3.1.

The position of (5242) is adjusted so that the two-dimensional frequencyrepresentation obtained is centered.

Step 35—Adjustment

of the focal length of the doublet (5151) (5152)

of the focal length of the doublet (5192)(5193)

of the assembly (5212) in rotation

of the CCDs (5198) and (5201) in rotation around the optical axis

The beams FRG, FED, FEGI are used. The mirror (5113) is put back inposition E (offcentered).

A program calculates the two-dimensional frequency representationreceived on the CCDs (5198) (5201) according to the procedure 9.3.1.

The positions of all the elements are adjusted so that:

the coordinates of the points associated with FRG, FED, FEGI arecorrect, i.e. with the notations used in 9.5.1.: (FRG,FEGI)=(FRD,FEDI)and (FEGI,FED)=(FEG,FEDI) in which the positions of the pointsFRD,FEG,FEDI are those which were obtained during Steps 24 and 28, allthese points being represented in FIGS. 78 and 79.

the images of FEP and FEG remain point images

More precisely:

The focal length of the doublet (5192)(5193) is adjusted so that thedistance between the points corresponding to FEGI and FED are correct.As the changes in the position of the doublet lenses can result in aloss of focus, the positions of the doublet lenses are also adjusted sothat the image of FED and FEGI remains a point image.

The rotation position of the CCDs (5198) and (5201) is adjusted so thatthe segment FEGI-FED is horizontal.

The focal length of the doublet (5151) (5152) and the position of theassembly (5151) (5152) are adjusted so that the distance between FRG andFEGI is correct and so that the line linking these points is vertical.As the changes in the position of the doublet lenses can result in aloss of focus, the positions of the doublet lenses are also adjusted sothat the image of FEGI remains a point image.

Following this adjustment, the mirror (5113) is brought back to positionC.

Steps 32 to 35 can be started again in this order a certain number oftimes to converge towards the correct positions. Step 35 in fact leadsto a loss of adjustment of the central position of the CCDs and of theorientation of the reference beams used in position C.

Step 36: Adjustment of the Position of Glass (5165) in the Direction ofthe Optical Axis

The beams FSI and FRD are used. A frequency image is obtained using theprocedure 9.4.1. from the sensors (5171) (5174). On this frequencyimage, the black spot of the glass can be seen dark against a lightbackground. The position of the glass is adjusted so that the black spotstands out against the background with the best possible contrast.

Step 37: Adjustment of the Position of Glass (5191) in the Direction ofthe Optical Axis

The beams FS and FRG are used. A frequency image is obtained by theprocedure 9.4. from the sensors (5201) (5198). On this frequency image,the black spot of the glass can be seen dark against a light background.The position of the glass is adjusted so that the black spot stands outagainst the background with the best possible contrast.

Step 38: Adjustment in Rotation Around the Optical Axis and Two-axisTranslation of the Diaphragms (5164) and (5190)

Prior to this adjustment, the mirror (5113) is brought back to themiddle position.

These diaphragms have the form indicated in FIG. 82. They are adjustablein translation on three axes and in rotation around the optical axis.The beams FS,FSI,FRG,FRGI are used for their adjustment. A frequencyimage is obtained by the procedure 9.4. from each of the sensors (5174)(5198).

They are adjusted in translation to have an image centered on eachsensor and as clear as possible without however hindering the movementof the mobile glass (5165) or (5191) associated with them. They areadjusted in rotation so that the coordinates of the point hidden by themobile part (5712) are the same on the image obtained from each sensor.These coordinates will hereafter be denoted (i_(o), j_(o)). Followingthis adjustment, the program also determines the radius R_(ouv) of theimage obtained on each sensor, which characterizes the zone accessibleby the beams not stopped by the aperture of the objective. It ispreferable to underestimate slightly R_(ouv).

Following this adjustment procedure of, the objectives are positioned sothat FS has a centered point image on each sensor. This adjustment willnot be modified until the introduction of the sample.

9.6. Beam deflector control

To each position of the mirror there corresponds a maximum (FE in FIG.75) of the frequency image obtained with the procedure 9.4. The point FRcorresponding to the reference wave is at the center of the image. Thepoint FO corresponding to the optical center is in the middle of FR andFE. The coordinates of the point FE in a coordinate system centered onFO are the equivalent of the coordinates of the point of direct impactof the reference wave relation to the optical center in the thirdembodiment.

The index p will characterize the right or left side of the microscopewhere p=0 for the sensors (5171)(5174) and p=1 for the sensors(5198)(5201).

The system for positioning the beam deflection mirror consists of twono-hysteresis actuators. For example, it is possible to use steppingmotor positioning as described in the first embodiment. If the system isdesigned with sufficient care, such a system can exhibit very littlehysteresis. The position of the mirror is then characterized by thenumber of steps completed by each motor from a central point.Piezoelectric positioners designed with a feedback loop allowing precisecontrol of their extension can also be used, in which case the number ofsteps of each motor is replaced by the extension of each actuator.Piezoelectric positioners without feedback can also be used; howeverthey exhibit a strong hysteresis which requires the determination ofvoltages used over a given path, a point-to-point determination of themirror position values, as carried out below, not being possible.

The control of the motors is defined by two arrays tab1 and tab2, wheretab1[p,i,j] ( resp. tab2[p,i,j]) is the number of steps actuator 1(resp.2) must complete so that the point FE has the coordinates (i,j) onthe image obtained by procedure 9.4. from a sensor indexed by p, in acoordinate system in which FO has the coordinates$\left( {\frac{N_{pix}}{2},\frac{N_{pix}}{2}} \right).$

Actuators are numbered so that actuator 1 determines a movement alongthe axis i and actuator 2 determines a movement along the axis j.

The determination of the arrays tab1 and tab2 is done without an object(transparent plate), using a specific program. The program thendetermines, for each “objective” point, the number of correspondingsteps of each motor from an origin. FIG. 77 represents an example of analgorithm of such a program. The main steps are:

(5407): Modification of shutter control. When p=0, the beam used must beFEG and, when p=1, the beam used must be FED.

(5401): imax and jmax correspond to the coordinates of the maximumobtained using procedure 9.4. from the sensor receiving the point ofdirect impact of the illuminating wave. We then have:$x = \frac{imax}{2}$ $y = \frac{jmax}{2}$

(5402): The value lim used depends on the accuracy of the actuators. Itis possible to have, for example, lim=0.25.

(5403): In a manner similar to what was done in 5.20., the displacementof the motors is equal to:

pas 1=(i−x).pas _(—) par_pixel/2

pas 2=(j−y).pas _(—) par_pixel/2

 The current position value is modified:

pos 1+=pas 1

pos 2−=pas 2

pas_par_pixel is the ratio (number of motor displacement steps)/(numberof pixels of the displacement of FE on the image obtained in 9.4.). Itmust have been previously determined, similar to what was done in thefirst embodiment, but with the images now calculated according to theprocedure 9.4. and not obtained directly on the CCD.

(5404): The displacement of the motors is carried out, for a number ofsteps pas1, pas2 of each motor.

(5405): The current values of the coordinates in motor steps arerecorded in an array:

tab 1[p,i,j]=pos 1

tab 2[p,i,j]=pos 2

 However, in the case of${\left( {i - \frac{N_{pix}}{2}} \right)^{2} + \left( {j - \frac{N_{pix}}{2}} \right)^{2}} > R_{ouv}^{2}$

 the values assigned are, for example, negative values indicating anerror, such as tab1[p,i,j]=−10000, tab2[p,i,j]=−10000

(5406): The motors are displaced by -pos1, -pos2 so as to return to theorigin, which must remain constant to avoid distorting the path.

To obtain an illuminating wave on the sensor p, at the point ofcoordinates i,j, the two motors will be displaced up to their positionscharacterized respectively by tab1[p,i,j] and tab2[p,i,j], and theshutters (5144) and (5207) will be actuated according to the followingarray:

Index p Position of (5144) Position of (5207) 0 closed open 1 openclosed

9.7. Control of lasses (5165) and (5191)

Each of these glasses is mounted for example on a two-axis translationpositioner controlled by stepping motors and allowing the displacementof the glass in a plane orthogonal to the optical axis. For eachposition of the illuminating wave, characterized by the indices p,i,j,the glass must be positioned to cancel the direct illumination. This isdone by controlling the stepping motors to move the glass up to aposition characterized in number of steps of each motor by thecoordinates tabv1[p,i,j], tabv2[p,i,j].

Prior to this use of the glasses, a program is used to determine thearrays tabv1[p,i,j], tabv2[p,i,j]. An example of such a program isdescribed in FIG. 77.

To use the program of FIG. 77, the beams FS,FSI,FRD and FRG arenecessary and the object used is a transparent plate. The filter (5128)used on the path of the beam FS is adjusted so that the sensors aresaturated on a few pixels around the points of impact of the beams FSand FSI.

The main steps in this program are:

(5107): The position of the mirror (5113) is controlled by the arraystab1[p,0,0], tab2[p,0,0].

(5401): An image is obtained using the procedure described in 9.4. withFourier transformation from a sensor located on the side characterizedby the index p. The modulus of the elements of this image is extractedto obtain an image in real numbers on which the point cut off by the“black spot” of the glass appears in a light tone against a blackbackground. The image thus obtained constitutes the array S ofdimensions 2N_(pix)×2N_(pix) from which the coordinates imax and jmax ofthe maximum are calculated using the procedure described in 7.3.3.1. Wethen have:

x=imax

y=jmax

(5402): The value rim used depends on the accuracy of the positioner. Itis possible to have, for example, lim=0.25.

(5403): Similar to what was done in 5.20., the displacement of themotors is equal to:

pas 1=(i−x).pas _(—) par_pixel/2

pas 2=(i−y).pas _(—) par_pixel/2

 The current position value is modified:

pos 1+=pas 1

pos 2+=pas 2

pas_par_pixel is the ratio (number of motor displacement steps)/(numberof pixels of the displacement of the point of coordinates imax,jmax onthe image obtained in 9.4.). It must have been previously determined,for example by displacing the glass along one of the directions andmeasuring the number of displacement pixels of the point of coordinates(imax,jmax).

(5404): The displacement of the motors is completed, for a number ofsteps pas1, pas2 of each motor.

(5405): The current values of the coordinates in motor steps arerecorded in a array:

tabv 1[p,i,j]=pos 1

tabv 2[p,i,j]=pos 2

 However, in the case of${\left( {i - \frac{N_{pix}}{2}} \right)^{2} + \left( {j - \frac{N_{pix}}{2}} \right)^{2}} > R_{ouv}^{2}$

 the values assigned are, for example, negative values indicating anerror, such as tab1[p,i,j]=−10000, tab2[p,i,j]=−10000

(5406): The motors are displaced by -pos1, -pos2 so as to return to theorigin, which must remain constant to prevent the distortion of thepath.

9.8. Obtaining the constant K

The micrometer is introduced as a sample. An image is obtained in thepresence of the beams FED and FRG using the procedure described in 9.4.in which Step 3 with its Fourier transformation is not performed. Themodulus of the complex values of this image is used to obtain a realimage of the intensity. The position of the sample is adjusted so thatthis image is correctly focussed. The distance in pixels D_(pix) betweentwo graduations separated by a real distance D_(reel) is measured. Thevalue of the constant K is${{then}\quad K} = {\frac{n_{v}}{\lambda}\quad \frac{2\quad N_{pix}}{D_{pix}}\quad {D_{reel}.}}$

9.9. Obtaining a complete two-dimensional frequency representation

In the third embodiment, a two-dimensional frequency representation wasobtained using the procedure described in 7.12. In the presentembodiment, the procedure for obtaining this representation must bemodified.

9.9.1. Principle

9.9.1.1. Acquisition

During acquisition, the index r will designate the open switch,according to the table below:

Index r Shutter (5109) Shutter (5104) 0 open closed 1 closed open

The index p will designate the right side (p=0) or left side (p=1) ofthe microscope where the direct illuminating wave arrives and,consequently, the position of the shutters (5144) and (5207), accordingto the table below:

Index p Shutter (5207) Shutter (5144) 0 open closed 1 closed open

The index q will designate the side of the microscope from which theacquired data come, with q=0 for data coming from the side where thedirect illuminating wave arrives and q=1 for the data coming from theopposite side.

We take s=p{overscore (q)}+{overscore (p)}q . s thus designates the sideof the microscope from which the acquired data come, with s=0 for theright side and s=1 for the left side.

We assign the indices (s,t) to the sensors as follows.

Index s 0 1 Index t 0 1 0 1 Sensor (5174) (5171) (5198) (5201)

The data MA[k,p][d,r,t][q,i,j] come from the sensor indexed by(p{overscore (q)}+{overscore (p)}q,t) For each quintuplet (k,p,q,r,t),we obtain, from a corresponding sensor, and from three acquisitionscorresponding to different phases of the reference wave, an image in theform of a two-dimensional array of complex numbers with dimensions of2N_(pix)×2N_(pix).

9.9.1.2. Chance to frequency representation

The image thus obtained is in spatial representation. A Fouriertransformation is carried out to obtain an image in frequencyrepresentation. The representation thus obtained is centered around apoint FR corresponding to the reference beam. It must be translated andlimited to obtain a representation with dimensions of N_(pix)×N_(pix)centered around the point FO corresponding to the optical center.

FIGS. 75 and 76 depict the representations obtained respectively for agiven index p on the sensors indexed by q=0 and q=1. In these figures,the point denoted FR is the central point with coordinates of (N_(pix),N_(pix)), corresponding to the frequency of the reference wave. Thepoint denoted FE corresponds to the frequency of the illuminating wave(q=0) or its opposite wave (q=1). The point denoted FO corresponds tothe optical center of the system, i.e. to the frequency of a wavepassing through the object observed in the direction of the opticalaxis. The contours (5301) and (5303) delimit the representation.

In the case of q=0, FR and FE are symmetrical in relation to FO. In thecase of q=1, FR and FE are symmetrical in relation to a horizontal linepassing through FO.

The limited and centered frequency representation is obtained from thesefigures by extracting the area with dimensions N_(pix)×N_(pix) centeredaround the point FO, limited by (5302) or (5304) in the figures.

9.9.1.3. Combination of different polarizations

We denote as D_(p,s,r,t) the complex value measured at a point C of afrequency representation obtained after the procedure described in9.9.1.2., from the sensor indexed by s,r and t when the direction of theilluminating wave is characterized by p.

We denote as {right arrow over (w)}_(s,r,t) the electric field vector ofthe reference wave on the sensor characterized by the indices s, r andt.

In FIG. 80, the different combinations of the indices s,r,t arerepresented in the form of a table. In each box of the table:

the neutral axis of the phase plate located ahead of the sensorconcerned is shown as a broken line.

the electric field vector of the reference wave before passing throughthe phase plate, directed along {right arrow over (i)} or {right arrowover (j)}, is shown as a solid line.

the electric field vector {right arrow over (w)}_(s,r,t) of thereference wave after passing through the phase plate and reflection onthe semi-transparent mirror which superimposes it on the beam comingfrom the object, directed along ±{right arrow over (u)} or +{right arrowover (v)}, is also shown as a solid line. If s=0, this vector is deducedfrom the electric field vector of the wave before passing through theplate by a symmetry whose axis is the neutral axis of the blade. If s=1,an additional vertical axis symmetry must be completed.

The values of {right arrow over (w)}_(s,r,t) are deduced from thisfigure. A formula grouping all the values obtained is, with a constantbias: {right arrow over(w)}_(s,r,t)=(−1)^(srt+{overscore (s)}{overscore ((rt))}){right arrowover (i)}+(−1)^({overscore (rt)}){right arrow over (j)}

The phase of the reference wave differs between the sensors (5171) and(5174). We denote as α_(s) the ratio characteristic of the intensity andphase shift between the sensors indexed respectively by (s, t=0) and (s,t=1).

The wave detected on a given sensor constitutes a projection of the wavereaching this sensor on an axis oriented like the electric field vectorof the reference wave. The unit vector orienting this axis will bedenoted as the electric field vector of the reference wave.

For the measurements will be used:

an illuminating wave directed along (−1)^(p){right arrow over (i)} wherep is the index of the sensor toward which the direct illuminating waveis directed. The factor (−1)^(p) is due to the fact that the horizontalcomponent of the illuminating wave, symmetrized by the mirrors whichdirect it in two opposite directions according to the index p, isreversed when the direction of the illuminating wave is itself reversed.For this illuminating wave, the components of the diffracted wavepolarized along the axes oriented by {right arrow over (w)}_(s,0,0) and{right arrow over (w)}_(s,0,1) will be measured, giving respectively thefactors D_(p,s,0,0) and D_(p,s,0,1)

an illuminating wave directed along {right arrow over (j)}. For thisilluminating wave, the components of the diffracted wave polarized alongthe axes oriented by {right arrow over (w)}_(s,1,0) and {right arrowover (w)}_(s,1,1) will be measured, giving respectively the factorsD_(p,s,1,0) and D_(p,s,1,1)

When the electric field vector of the illuminating beam (at the point E)is A₀(−1)^(p){right arrow over (i)}+A₁{right arrow over (j)}, theelectric field vector resulting at the point C, on side s of themicroscope, is thus:${\sum\limits_{r,t}}\quad A_{r}\quad D_{p,s,r,t}\quad \left( {\overset{\_}{t} + {t\quad \alpha_{s}}} \right)\quad {\overset{\rightharpoonup}{w}}_{s,r,t}$

When the electric field vector at the point E is A₀{right arrow over(i)}+A₁{right arrow over (j)}, the electric field vector resulting atpoint C, on side s of the microscope, is thus:${\sum\limits_{r,t}}\quad {A_{r}\quad \left( {- 1} \right)^{p\overset{\_}{r}}\quad D_{p,s,r,t}\quad \left( {\overset{\_}{t} + {t\quad \alpha_{s}}} \right)\quad {\overset{\rightharpoonup}{w}}_{s,r,t}\quad {or}\quad {{\sum\limits_{r,t}}\quad \left( {- 1} \right)^{p\overset{\_}{r}}\quad \left( {{\left( {- 1} \right)^{{srt} + {\overset{\_}{s}\quad \overset{\_}{({rt})}}}\overset{\rightharpoonup}{i}} + {\left( {- 1} \right)^{\overset{\_}{r}\quad \overset{\_}{t}}\quad \overset{\rightharpoonup}{j}}} \right)\quad \left( {\overset{\_}{t} + {t\quad \alpha_{s}}} \right)\quad A_{r}\quad D_{p,s,r,t}}}$

which corresponds to the expression

C _(p,s,0,0) A ₀ {right arrow over (i)}+C _(p,s,1,0) A ₀ {right arrowover (j)}+C _(p,s,0,1) A ₁ {right arrow over (i)}+C _(p,s,1,1) A ₁{right arrow over (i)}+C _(p,s,1,1) A ₁ {right arrow over (j)}

where:$C_{p,s,d,r} = {{\sum\limits_{t}}\quad \left( {- 1} \right)^{p\overset{\_}{r}}\quad \left( {{\left( {- 1} \right)^{{srt} + {\overset{\_}{s}\quad \overset{\_}{({rt})}}}\quad \overset{\_}{d}} + {\left( {- 1} \right)^{\overset{\_}{r}\quad \overset{\_}{t}}d}} \right)\quad \left( {\overset{\_}{t} + {t\quad \alpha_{s}}} \right)\quad D_{p,s,r,t}}$

or:

C _(p,s,0,0)=(−1)^(p)(−1)^({overscore (s)})(D _(p,s,0,0)+α_(s) D_(p,s,0,1))

C _(p,s,1,0)=(−1)^(p)(−D _(p,s,0,0)+α_(s) D _(p,s,0,1))

C _(p,s,0,1)=(−1)^(s)(−D _(p,s,1,0)+α_(s) D _(p,s,1,1))

C _(p,s,1,1) =D _(p,s,1,0)+α_(s) D _(p,s,1,1)

The same expression as in 7.12 can be used from these values ofC_(p,s,d,r) to obtain the value sought M_(p,s), the indices p,s whichwere not necessary in 7.12. being added:

M _(p,s)=−cos φ_(e) cos φ_(c) C _(p,s,0,0)−sin φ_(e) cos φ_(c) C_(p,s,0,1)−cos φ_(e) sin φ_(c) C _(p,s,1,0)−sin φ_(e) sin φ_(c) C_(p,s,1,1)

${{or}\quad M_{p,s}} = {{\sum\limits_{r,t}}\quad E_{p,s,r,t}\quad D_{p,s,r,t}}$

where:

E _(p,s,0,0)=(−1)^(p)((−1)^(s) cos φ_(e) cos φ_(c)+cos φ_(e) sin φ_(c))

E _(p,s,0,1)=(−1)^(p)α_(s)((−1)^(s) cos φ_(e) cos φ_(c)−cos φ_(e) sinφ_(c))

E_(p,s,1,0)=(−1)^(s) sin φ_(e) cos φ_(c)−sin φ_(e) sin φ_(c)

 E _(p,s,1.1)=α_(s)(−(−1)^(s) sin φ_(e) cos φ_(c)−sin φ_(e) sin φ_(c))

It is necessary to first determine the coefficient α_(s). Around thepoint of impact of the illuminating wave, a comparable value is obtainedon each sensor for a given index r. It is thus possible to adopt forthis coefficient the value$\alpha_{s} = \frac{\sum{D_{p,s,r,0}\overset{\_}{D_{p,s,r,1}}}}{\sum\left| D_{p,s,r,1} \right|^{2}}$

where the sums are on the indices r and on a reduced number of pointsaround the point of direct impact of the illuminating wave.

9.9.2. Acquisition

As in the third embodiment, the point of direct impact of theilluminating wave follows a path indexed by k and characterized by thearrays Io[k],Jo[k].

As in 7.12.2.1., the acquisition of the elementary images is aniteration on the integers p and k designating respectively the sensorreached by the direct illuminating beam and the order number of theelementary image in the series of images corresponding to a given indexp.

There is no beam attenuation here. For each pair (k,p) complying with${{\left( {{{Io}\lbrack k\rbrack} - \frac{N_{pix}}{2}} \right)^{2} + \left( {{{Jo}\lbrack k\rbrack} - \frac{N_{pix}}{2}} \right)^{2}} \leq R_{ouv}^{2}},$

the program controls the displacement of the mirror through the arraystab1[p,Io[k],Jo[k]], tab2[p,Io[k],Jo[k]] and the shutters (5144) and(5207) as indicated in 9.6. It moves the glass located on the sideopposite the point of direct impact of the illuminating wave so that itsblack spot is outside the frequency field used.

It controls the movement of the glass located on the side where thepoint of direct impact of the illuminating wave is located as indicatedin 9.7., with the arrays tabv1[p,Io[k],Jo[k]], tabv2[p,Io[k],Jo[k]].

However, according to a variant which will be called Variant 2, it doesnot use this glass and hence moves it so that its black spot is outsidethe frequency field used.

To increase the speed, the number of changes of the index p must belimited as much as possible, calling for a manipulation of the shutters(5144) and (5207) which are slower than (5104) and (5109) for example.The program thus carries out a first iteration on k, at p=0, followed bya second iteration on k, at p=1.

For each value of (k,p) the program acquires 12 pairs of elementaryimages. A pair of elementary images is, as in 7.12.2.1., an arrayindexed

on the one hand, by the index q with q=0 if the image is detected on thesame side as the point of direct impact of the reference wave, and q=1if it is detected on the opposite site.

on the other hand, by the indices i and j characterizing the position ofthe pixel on the sensor concerned.

The indices p,q,r,t are defined as indicated in 9.9.1.1. The index ddetermines the phase shift of the reference wave and is defined as inthe following table:

Phase shift θ_(d) Index d (degrees) 0 −120 1 0 2 +120

The acquisition of a series of images corresponding to the indicesk,p,q,d,r,t thus generates the series of elementary image pairsMA[k,p][d,r,t][q,i,j]

The program also acquires a series of images corresponding to thereference image. It moves the motors up to the positionspos1[p,i_(r),j_(r)] pos2[p,i_(r),j_(r)] where (i_(r),j_(r)) are normallythe coordinates (i_(o),j_(o)) of the fixed point hidden by thediaphragrms, determined in Step 38 of the adjustment procedure 9.5.However, if Variant 2 is used, (i_(r),j_(r)) are the coordinates ofanother fixed point, highly offcentered but not blocked. The programthen acquires a series of six pairs of elementary images, obtaining anarray MA2[k,p][d,r,t][q,i,j]

However, according to a variant which will be called Variant 1, theprogram does not carry out the acquisition of this reference image.

9.9.3. Calculations

The calculation of the two-dimensional frequency representation, of thereference image, and of the characteristics arrays of the noise on thesetwo images is carried out by Steps 1 to 8 below:

Step 1—Calculation of Two-dimensional Spatial Representations

The program performs, on all the data obtained:${{{{MB}\left\lbrack {k,p} \right\rbrack}\left\lbrack {r,t} \right\rbrack}\left\lbrack {q,i,j} \right\rbrack} = {{\frac{1}{6}\quad \left( {{2\quad {{{{MA}\left\lbrack {k,p} \right\rbrack}\left\lbrack {0,r,t} \right\rbrack}\left\lbrack {q,i,j} \right\rbrack}} - {{{{MA}\left\lbrack {k,p} \right\rbrack}\left\lbrack {1,r,t} \right\rbrack}\left\lbrack {q,i,j} \right\rbrack} - {{{{MA}\left\lbrack {k,p} \right\rbrack}\left\lbrack {2,r,t} \right\rbrack}\left\lbrack {q,i,j} \right\rbrack}} \right)} + {\overset{\sim}{j}\quad \frac{1}{2\sqrt{3}}\quad \left( {{{{{MA}\left\lbrack {k,p} \right\rbrack}\left\lbrack {1,r,t} \right\rbrack}\left\lbrack {q,i,j} \right\rbrack} - {{{{MA}\left\lbrack {k,p} \right\rbrack}\left\lbrack {2,r,t} \right\rbrack}\left\lbrack {q,i,j} \right\rbrack}} \right)}}$

Step 2—Change to Frequency Representation

The indices i and j vary from 0 to 2 N_(pix)−1. The program performs theFourier transform, based on these two indices, of each of the spatialrepresentations previously obtained. This leads to the transformedrepresentations:

MC[k,p][r,t][q,i,j]

Step 3: Compensation for Frequency Response of the Sensors

This compensation is not indispensable, but it improves substantiallythe precision of the microscope.

The program calculates the array MD:

MD[k,p][r,t][q,i,j]=RI[i,j]MC[k,p][r,t][q,i,j]

The array RI represents the inverted frequency response of the sensors.It is determined in 9.11. However, according to a variant which will becalled Variant 3, used for certain adjustments, the array RI is set to1.

Step 4—Translation and Limitation of the Two-dimensional FrequencyRepresentation

The position of the point FO indicated in 9.9.1.2. is stored in the formof arrays Io[k] Jo[k]. The considerations indicated in 9.9.1.2. thenresult in the following operations performed by the program:

ME[k,p][r,t][0,i,j]=MD[k,p][r,t][0,i+Io[k],j+Jo[k]]

ME[k,p][r,t][1,i,j]=MD[k,p][r,t][1,i−Io[k]+N _(pix) ,j+Jo[k]]

where i and j vary now from 0 to N_(pix)−1.

Step 5: Calculation of Coefficients α_(s)

A coefficient α_(s) is determined for each triplet (k,p,q). It is storedin an array of complex numbers alpha[k,p,q]. The program goes throughall the triplets (k,p,q). For each triplet

it initializes to 0 the numbers nom and denom.

it goes through all the triplets r,i,j testing the condition:

(i−Io[k]) ²+(j−Jo[k]) ²≦lim²

where, for example, lim=20.

 When the condition is true, it performs:

nom+=ME[k,p][r,0][q,i,j]{overscore (ME[k,p][r,1][q,i,j])}

denom+=|ME[k,p][r,1][q,i,j]| ²

When the loop on r,i,j is completed, it performs:

alpha[k,p,q]=nom/denom

Step 6—Combination of Values Corresponding to Indices r and t

This step is designed to calculate M_(k,p,q)[i,j] as a function ofME[k,p][r,t][q,i,j]. This can be accomplished very simply by performingthe following operation: M_(k,p,q)[i,j]=ME[k,p][1,1][q,i,j], the valueassigned to the noise then being constant and Step 5 becomesunnecessary. If this method is to be used, it is however preferable toeliminate the phase plates (5111) (5238) (5239) (5248) (5249).

In any case, this method, similar to the one used in the secondembodiment, induces imperfections in the high frequencies. It is thuspreferable to use the method whose principle was indicated in 9.9.1.3.,the quantity ME[k,p][r,t][q,i,j] being the measured value denotedD_(p,s,r,t) in 9.9.1.3. Step 6 is then carried out as follows:

For each value of the indices k,p,q,i,j, the program calculates:${x_{c} = \frac{j - \frac{N_{pix}}{2}}{\frac{n_{o}}{n_{v}}\quad K}},{y_{c} = \frac{j - \frac{N_{pix}}{2}}{\frac{n_{o}}{n_{v}}\quad K}},{z_{c} = \sqrt{1 - x_{c}^{2} - y_{c}^{2}}}$${x_{e} = \frac{{{Io}\lbrack k\rbrack} - \frac{N_{pix}}{2}}{\frac{n_{o}}{n_{v}}\quad K}},{y_{e} = \frac{{{Jo}\lbrack k\rbrack} - \frac{N_{pix}}{2}}{\frac{n_{o}}{n_{v}}\quad K}},{z_{e} = \sqrt{1 - x_{e}^{2} - y_{e}^{2}}}$

 V _(yz) =y _(c) z _(e) −z _(c) y _(e)

V _(xz) =−x _(c) z _(e) +z _(c) x _(e)

V _(xy) =−x _(c) y _(e) +y _(c) x _(e)

M _(c) ^(c) =x _(c) ² +y _(c) ²

M _(e) ² =x _(e) ² +y _(e) ²

M _(ce) ={square root over (V_(yz) ²+V_(xz) ²+V_(xy) ²)}

The values of sin φ_(c) cos φ_(c) sin φ_(e) cos φ_(e) are determinedaccording to the following tables:

M_(ce) 0 other M_(c) 0 other cosφ_(c) 1 $- \frac{y_{e}}{M_{e}}$

$\frac{1}{M_{c}^{2}M_{ce}}\left( {{y_{c}^{2}V_{yz}} - {x_{c}y_{c}V_{xz}} + {x_{c}V_{xy}}} \right)$

sinφ_(c) 0 $\frac{x_{e}}{M_{e}}$

$\frac{1}{M_{c}^{2}M_{ce}}\left( {{{- x_{c}}y_{c}V_{yz}} + {x_{c}^{2}V_{xz}} + {y_{c}V_{xy}}} \right)$

M_(ce) 0 other M_(e) 0 other cosφ_(e) −1  $- \frac{y_{c}}{M_{c}}$

${- \frac{1}{M_{e}^{2}M_{ce}}}\left( {{y_{e}^{2}V_{yz}} - {x_{e}y_{e}V_{xz}} + {x_{e}V_{xy}}} \right)$

sinφ_(e) 0 $\frac{x_{c}}{M_{c}}$

${- \frac{1}{M_{e}^{2}M_{ce}}}\left( {{{- x_{e}}y_{e}V_{yz}} + {x_{e}^{2}V_{xz}} + {y_{e}V_{xy}}} \right)$

then the coefficients are calculated:

 coef[k,p,q,i,j][0,0]=(−1)^(p)((−1)^(p{overscore (q)}+{overscore (p)}q)cos φ_(e) cos φ_(c)+cos φ_(e) sin φ_(c))

coef[k,p,q,i,j][0,1]=alpha[k,p,q](−1)^(p)((−1)^(p{overscore (q)}+{overscore (p)}q)cos φ_(e) cos φ_(c)−cos φ_(e) sin φ_(c))

coef[k,p,q,i,j][1,0]=(−1)^(p{overscore (q)}+{overscore (p)}q) sin φ_(e)cos φ_(c)−sin φ_(e) sin φ_(c)

coef[k,p,q,i,j][1,1]=−alpha[k,p,q](−(−1)^(p{overscore (q)}+{overscore (p)}q)sin φ_(e) cos φ_(c)−sin φ_(e) sin φ_(c))

These coefficients do not depend on the result of the imaging operationsand, if the same series is always repeated, they can be stored in anarray instead of being recalculated each time. The program then usesthese values to combine the images obtained with the different positionsof the rotators as follows:${M_{k,p,q}\left\lbrack {i,j} \right\rbrack} = {\sum\limits_{r,t}^{\quad}\quad {{{{{ME}\left\lbrack {k,p} \right\rbrack}\left\lbrack {r,t} \right\rbrack}\left\lbrack {q,i,j} \right\rbrack}\quad {{{coef}\left\lbrack {k,p,q,i,j} \right\rbrack}\left\lbrack {r,t} \right\rbrack}}}$

coef[k,p,q,i,j][r,t] corresponds to the quantity denoted E_(p,s,r,t) en9.9.1.3. with s={overscore (p)}q+p{overscore (q)}.

ME[k,p][r,t][q,i,j] corresponds to the quantity denoted D_(p,s,r,t) in9.9.1.3.

M_(k,p,q)[i,j] to the quantity denoted M_(p,s) in 9.9.1.3.

Step 7: The Noise Amplitude is Calculated as Follows:

B_(k,p,0)[i,j] is equal to:

when RI[i+Io[k],j+Jo[k]]≠0: B_(k,p,0)[i,j]=|RI[i+Io[k],j+Jo[k]]|

otherwise: B_(k,p,0)[i,j]=MAX where MAX is a high value, for example10¹⁰

B_(k,p,1)[i,j] is equal to:

when RI[i−Io[k]+N_(pix),j+Jo[k]]≠0:B_(k,p,1)[i,j]=|RI[i−Io[k]+N_(pix),j+Jo[k]]|

otherwise: B_(k,p,1)[i,j]=MAX

Step 8—Calculation of the Reference Image

The reference image is calculated exactly in the same manner as the mainimage, but using array MA2 instead of array MA and replacing the valuesIo[k],Jo[k] by (i_(r),j_(r)). We shall denote as H_(k,p,q)[i,j] thearray thus generated and BH_(k,p,q)[i,j] the corresponding noiseamplitude. However, in the case of Variant 1, this reference image isnot calculated.

9.10. Fine adjustment of the position of the sensors

The purpose of this operation is to ensure a precise adjustment, inrelation to each other, of the two sensors corresponding to the sameside of the microscope, in particular in the direction of the opticalaxis. In the absence of such an adjustment, the origin of the frequencyrepresentations generated from each sensor may differ slightly, whichwould compromise subsequently the proper superposition of the portionsof the frequency representations coming from each side of themicroscope. Omitting this adjustment does not prevent the generation ofthree-dimensional representations of the sample, but limits theprecision of these representations.

A translation displacement of a sensor entails a modulation in thefrequency domain and hence a phase difference in the values of the planewaves obtained from this sensor. The adjustment consists in verifyingthat the plane waves received on each sensor for different directions ofthe illuminating beam are in phase.

The sensors are indexed with the indices s, t with the same conventionas in 9.9.1.3. The program initializes to 0 the arrays MF_(s,t) withdimensions of N_(pix)×N_(pix). It carries out a loop on the indices s, iand j. For each triplet s,i,j complying with${{\left( {i - \frac{N_{pix}}{2}} \right)^{2} + \left( {j - \frac{N_{pix}}{2}} \right)^{2}} \leq R_{ouv}^{2}},$

it performs the following operations:

it actuates the shutters so as to use the beam FEG for s=0 and the beamFED for s=1.

it moves the mirror to the point corresponding to tab1[s,i,j],tab2[s,i,j]

it generates, from each sensor indexed by s, a two-dimensional imageaccording to the method described in 9.4., thus obtaining two imagesMH_(s,t)[k,l] where the indices s and t have been added to characterizethe sensor, with the same convention as in 9.9.

it moves the mirror to a fixed point of coordinates (i_(r),j_(r))

it generates, from each sensor indexed by s, a two-dimensional imageaccording to the method described in 9.4., obtaining two imagesMHR_(s,t)[k,l].

it carries out, for the pair of integers (i,j):${{MF}_{s,t}\left\lbrack {i,j} \right\rbrack} = \frac{{MH}_{s,t}\left\lbrack {{2i},{2j}} \right\rbrack}{{MHR}_{s,t}\left\lbrack {{2i},{2j}} \right\rbrack}$

After having completed this loop on s,i,j, the program calculates thefollowing deviations:$\sigma_{s} = {\sum\limits_{i,j}^{\quad}\quad {{{{MF}_{s,0}\left\lbrack {i,j} \right\rbrack} - {{MF}_{s,1}\left\lbrack {i,j} \right\rbrack}}}^{2}}$

The translation position of the sensor (5171) must be adjusted so as tominimize σ₀ and the position of the sensor (5201) must be adjusted so asto minimize σ₁. The program must thus loop on the calculation of thesedeviations until the adjustment is completed. σ_(s) represents thestandard (rms) deviation due to the phase errors between the frequenciesreceived on the two sensors located on the same side of the microscope.

9.11. Determining the frequency response of the sensors

The sensors filter the spatial frequencies reaching them. This filteringis in the best case equivalent to an averaging on the surface of thepixel; however, in general, it is more significant owing to the defectsof the CCDs and of the cameras. In the embodiments based on a receptionin a frequency plane, this filtering simply induces a darkening of thepart of the generated image which is far from the center. In thisembodiment, this filtering is carried out in the spatial plane and posesmore significant problems. It is preferable to use cameras of goodquality and to compensate for this filtering. This filtering in thespatial plane is equivalent to a multiplication in frequency by a arrayRF whose inverse RI is used in the procedure 9.9. to obtain thecompensation. To limit the noise on the frequency response, the latteris acquired by means of plane waves rather than from a single sphericalwave.

9.11.1. Acquisition

The arrays Io and Jo are used, characterizing a path across all theaccessible points, i.e. (Io[k],Jo[k]) must go through all points thatcomply with${\left( {{{Io}\lbrack k\rbrack} - \frac{N_{pix}}{2}} \right)^{2} + \left( {{{Jo}\lbrack k\rbrack} - \frac{N_{pix}}{2}} \right)^{2}} \leq R_{ouv}^{2}$

where R_(ouv) is the radius limited by the aperture of the objectives,which was obtained in Step 37 of the adjustment procedure 9.5. This pathwill be called “complete path” hereafter.

A program carries out the acquisition defined by these arrays, accordingto the procedure described in 9.9. with the Variants 2 and 3, i.e.without using the glasses (5165)(5191) and with a point of coordinates(i_(r),j_(r)) not blocked. It thus generates the series of imagesM_(k,p,q)[i,j] and H_(k,p,q)[i,j]. The index n_(o) not being known, itis taken equal to n_(v) in the procedure 9.9. However, during thisacquisition, it is sufficient to record the valuesM_(k,p,q)[Io[k],Jo[k]] and H_(k,p,q)[i_(r),j_(r)].

9.11.2. Calculation

The program initializes to 0 the array RF1 of dimensionsN_(pix)×N_(pix). It then goes through the series of indices k. For eachindex k, it performs:${{RF1}\left\lbrack {{{Io}\lbrack k\rbrack},{{Jo}\lbrack k\rbrack}} \right\rbrack} = {{\frac{M_{k,0,0}\left\lbrack {{{Io}\lbrack k\rbrack},{{Jo}\lbrack k\rbrack}} \right\rbrack}{H_{k,0,0}\left\lbrack {i_{r},j_{r}} \right\rbrack}} + {\frac{M_{k,1,0}\left\lbrack {{{Io}\lbrack k\rbrack},{{Jo}\lbrack k\rbrack}} \right\rbrack}{H_{k,1,0}\left\lbrack {i_{r},j_{r}} \right\rbrack}}}$

where i_(r),j_(r) are the coordinates of the maximum of the referenceimage, as defined in 9.9.

It then takes the two-dimensional inverse Fourier transform of the arrayRF1, obtaining an array RF2. It completes this array with zeros byinitializing to 0 the array RF3 of dimensions 2N_(pix)×2N_(pix) and thenby carrying tout for i and j varying from 0 to N_(pix)−1:${{RF3}\left\lbrack {{i + \frac{N_{pix}}{2}},{j + \frac{N_{pix}}{2}}} \right\rbrack} = {{RF2}\left\lbrack {i,j} \right\rbrack}$

It then takes the two-dimensional inverse Fourier transform of the arrayRF3, obtaining the array RF corresponding to the frequency response ofthe sensors.

The program then calculates the inverse frequency response of thesensors as follows:

if (i−N_(pix))²+(j−N_(pix))²≦4R_(ouv) ² then:${{RI}\left\lbrack {i,j} \right\rbrack} = \frac{1}{{{RF}\left\lbrack {i,j} \right\rbrack}}$

otherwise: RI[i,j]=0

The array RI thus determined is used in particular in 9.9.

9.12. Determining the relative coordinates of the origins of imagesobtained on each side of the microscope

The frequency images obtained on each side of the microscope byprocedure 9.9. are equivalent to the frequency images which wereobtained in the preceding embodiments. They are, in like manner,relative to an origin, which no longer depends on the origin of thereference wave, but on the position of the sensor. To be able tosuperimpose these representations, it is necessary to know the relativeposition of the origins of the representations obtained on each side ofthe microscope.

This is accomplished by using the spherical waves FS and FSI. Thespherical wave FS or FSI received on each side of the microscope must inprinciple be a point wave centered on each receiving surface. Theposition of the objectives can thus be adjusted in the presence of thebeams FS, FSI,FRG,FRD so that the image, obtained on each sensor by theprocedure 9.4. used without Fourier transformation, is a perfectcentered point image. If this adjustment of the position of theobjectives is carried out with great care and with sufficiently precisepositioners, offering sub-micrometric precision, for example microscopefocussing devices of sufficient quality, or positioners withpiezoelectric control, then the origins of the images obtained on eachside of the microscope coincide.

The adjustment procedure described in 9.5. however makes it possible toobtain this adjustment quality using positioners of average precisionfor the objectives. In fact, fine adjustments are carried out, in thisprocedure, by the displacement of the cameras and not of the objectives.If the adjustment procedure described in 9.5. is carried out carefully,the origins of the frequency representations finally obtained willcoincide. Their relative coordinates are thus (x,y,z)=(0,0,0).

However, the adjustments obtained with the procedure 9.5. or by anotherposition adjustment of the objectives are generally not perfect. Inparticular, they may be influenced by local imperfections of thesensors. It is possible to use (x,y,z)=(0,0,0), but a bettersuperposition of the images coming from each sensor will be obtained ifa suitable algorithm is used to calculate a more precise value of theseparameters. A precise determination of the relative positions of theorigins can be obtained using a method similar to the one used in 7.9.1.However:

The image of the beam FS or FSI in the receiving plane is a point imageand is not distributed over the entire receiving surface as in 7.9.1.The consequences are that the image obtained is sensitive to localdefects of the sensors and is very noisy.

The point of focus of FS and FSI does not correspond to thecharacteristic point (with the terminology used in 3.16.) of one of theobjectives, whereas in 7.9.1. the point of focus of the reference beamcorresponded to the characteristic point of a corresponding objective.

The method used is thus modified to overcome, to the extent possible,these drawbacks. The beams used are FS,FSI,FRD,FRG and the object usedis a transparent plate. The method used is broken down into two phases:

Acquisition phase: This phase consists of an iteration on the indicesk,l going through the set E0 of the points complying with${\left( {k - \frac{N_{pix}}{2}} \right)^{2} + \left( {l - \frac{N_{pix}}{2}} \right)^{2}} \leq {rayon}^{2}$

 where for example rayon=0.8R_(ouv). The array M6 of dimensionsN_(pix)×N_(pix) is initialized to 0 and then the program performs, foreach pair (k,l) of the set E0, Steps 1 to 7 below:

Step 1: Positioning of the beam deflection mirror at the pointdetermined by tab1[0,k,l], tab2[0,k,l]

Step 2: An image is acquired on each side of the microscope according tothe procedure described in 9.4. without performing the Fouriertransform. These images will be denoted M0 _(s)[i,j] instead of MH[i,j],the index s characterizing the sensor with s=0 for (5174) and s=1 for(5198).

Step 3: The program determines the coordinates imax_(s),jmax_(s) of themaximum value of each array M0 _(s)[i,j]. It then calculates the arraysM1 _(s)[i,j] with:

if (i−imax_(s))²+(j−jmax_(s))²≦R_(niv) ² M1 _(s)[i,j]=M0 _(s)[i,j]

otherwise M1 _(s)[i,j]=0

where R_(niv) is determined so that the disc of radius R_(niv) centeredon imax_(s),jmax_(s) contains all the points whose values without noiseare higher than the noise level, while remaining as small as possible.In practice, R_(niv) can be determined empirically and may be equal toten or so pixels.

Step 4: The program performs the two-dimensional Fourier transform ofeach array M1 _(s)[i,j], obtaining the arrays M2 _(s)[i,j].

Step 4: The program applies the filter RI to the arrays thus obtained:

M 3 _(s) [i,j]=M 2 _(s) [i,j]RI[i,j]

Step 5: The program calculates the array M4 of dimensionsN_(pix)×N_(pix) as follows:${{{\text{- if}\left( {i - \frac{N_{pix}}{2}} \right)^{2}} + \left( {j - \frac{N_{pix}}{2}} \right)^{2}} \leq {R_{ouv}^{2}\quad {then}}}\quad$${M_{4}\left\lbrack {i,j} \right\rbrack} = \frac{{M3}_{0}\left\lbrack {{i + k},{j + l}} \right\rbrack}{{M3}_{1}\left\lbrack {{i - k + N_{pix}},{j + l}} \right\rbrack}$

otherwise, M4[i,j]=0

 The array M4 represents the phase shift between the two sensors due tothe non-coincidence of the origins, partially noise corrected.

Step 6: The program performs the inverse Fourier transform of the arrayM4, obtaining an array M5.

Step 7: The program modifies the array M6 as follows:${{M6}\left\lbrack {i,j} \right\rbrack}+=\frac{{M4}\left\lbrack {i,j} \right\rbrack}{{M4}\left\lbrack {0,0} \right\rbrack}$

Calculation phase: The program calculates the array M7 which is theFourier transform of the array M6. The program calculates thecoordinates x,y,z obtained by the procedure described in 7.8. from thearray M7, which replaces the array denoted F_(rec) in 7.8. However, forthis step, the procedure described in 7.8. must be modified as indicatedin 7.9.1. to take into account the fact that the index of the object isknown. It must also be modified in a second manner to take into accountthe fact that the acquisition method is different and that the standarddeviation σ² must be calculated in the spatial domain and not in thefrequency domain as in 7.8.1. This second modification consists inreplacing the steps (3512) to (3514) in FIG. 50 by the following steps,which are carried out in spatial representation and in which it is thusthe array M6 (and not M7) which is used:

Step 1: The program determines the coordinates (imaxjmax) of the pointcorresponding to the maximum value of |M6[i,j]|. It then calculates${rapport} = \frac{{M6}\left\lbrack {{imax},{jmax}} \right\rbrack}{F_{d}\left\lbrack {{imax},{jmax}} \right\rbrack}$

Step 2: The program calculates the characteristic value max:$\max = {- {\sum\limits_{{({i,j})} \in {Disque}}\quad {{{{M6}\left\lbrack {i,j} \right\rbrack} - {{rapport} \cdot {F_{d}\left\lbrack {i,j} \right\rbrack}}}}^{2}}}$

where the sum is over all the pairs (i,j) included in the disc of center(imax,jmax) and of radius R_(niv)/2.

9.13. Determininu the phases of the beams

This procedure is similar to the procedure 7.9.2. The position of theobjectives is the one which was used in 9.12. to obtain the coordinatesx,y,z and must not be modified during the present procedure.

We use the arrays Io and Jo already used in 9.11. and characterizing acomplete path.

A program carries out the acquisition defined by these arrays, accordingto the procedure described in 9.9. with Variants 2 and 3, i.e. withoutusing the glasses (5165)(5191), with a point of coordinates(i_(r),j_(r)) not blocked, and without sensor filtering compensation. Itthus generates the series of images M_(k,p,q)[i,j] and H_(k,p,q)[i,j].The index n_(o), not being known, it is taken equal to n_(v) in theprocedure 9.9. During this acquisition, it is sufficient to record thevalues M_(k,p,q)[Io[k],Jo[k]] and H_(k,p,q)[i_(r),j_(r)].

The program initializes to 0 the array Ra and then goes through theseries of the indices k,p carrying out, for each pair k,p:${{Ra}\left\lbrack {p,{{Io}\lbrack k\rbrack},{{Jo}\lbrack k\rbrack}} \right\rbrack} = {\frac{M_{k,p,0}\left\lbrack {{{Io}\lbrack k\rbrack},{{Jo}\lbrack k\rbrack}} \right\rbrack}{H_{k,p,0}\left\lbrack {i_{r},j_{r}} \right\rbrack}{\exp \left( {{- \overset{\sim}{j}}\frac{2\pi}{\lambda}{n_{v}\left( {{x\frac{{Io}\lbrack k\rbrack}{K}} + {y\frac{{Jo}\lbrack k\rbrack}{K}} + {z\sqrt{1 - \frac{{{Io}\lbrack k\rbrack}^{2} + {{Jo}\lbrack k\rbrack}^{2}}{K}}}} \right)}} \right)}}$

where i_(r),i_(r) are the coordinates of the maximum of the referenceimage, as defined in 9.9. and where x,y,z are the coordinates determinedin 9.12.

9.14. Adiusting the position of the objectives in the presence of thesample

The sample to be observed is put in place. The beams FS and FRG areused. The beam FRG used is in the direction of the optical axis, so thatthe deflection mirror is in the central position, defined by${{tab1}\left\lbrack {0,\frac{N_{pix}}{2},\frac{N_{pix}}{2}} \right\rbrack},{{{tab2}\left\lbrack {0,\frac{N_{pix}}{2},\frac{N_{pix}}{2}} \right\rbrack}.}$

From the wave received on the sensor (5198), two images are generates bythe procedure described in 9.4.: a spatial image obtained withoutcarrying out Step 3 and a frequency image obtained by carrying out Step3. The modulus of the complex values is extracted on each image. Theposition of the objectives is adjusted on the same principle as in7.10.:

The spatial image must be centered.

On the frequency image, one should observe a light-toned disc. Theadjustment must be carried out so that the intensity is as high aspossible for the high frequencies (points far from the center). Theobserved disc must remain relatively uniform.

If a dark ring appears between the outer edge and the middle zone, thesample is too thick and all the frequencies cannot be taken intoaccount. The adjustment must then be carried out so as to have arelatively uniform disc with as high a radius as possible. The disc doesnot reach its maximum size and the high frequencies cannot be taken intoaccount. The resolution of the image, mainly in depth, is reduced. Theonly solution to this problem consists in using a specially designedobjective, described in Paragraph 7.19.

9.15. Determining x,y,z,L,n₀

This step is similar to the one described in procedure 7.11. As in theprocedure described in 7.11., this step can be avoided by the priormeasurement of the values L and n₀ and by introducing the sample withoutmoving the objectives, and hence without carrying out Step 9.14., so asnot to modify the values of x,y,z obtained in 9.12.

The program carries out the series of acquisitions defined by the arraysIo and Jo defining a complete path, already used in 9.11.2, according tothe procedure described in 9.9. used with Variants 2 and 3. It thusgenerates the series of images M_(k,p,q)[i,j] and H_(k,p,q)[i,j].However, during this acquisition, it is sufficient to record the valuesM_(k,p,q)[Io[k],Jo[k]] and H_(k,p,q)[i_(r),j_(r)].

The program then goes through the series of indices k. For each value ofk, it performs:${F_{rec}\left\lbrack {{{Io}\lbrack k\rbrack},{{Jo}\lbrack k\rbrack}} \right\rbrack} = {\frac{M_{k,0,0}\left\lbrack {{{Io}\lbrack k\rbrack},{{Jo}\lbrack k\rbrack}} \right\rbrack}{H_{k,0,0}\left\lbrack {i_{r},j_{r}} \right\rbrack}\frac{1}{{Ra}\left\lbrack {0,{{Io}\lbrack k\rbrack},{{Jo}\lbrack k\rbrack}} \right\rbrack}}$

The program described in 7.8. is then used to calculate the parametersx,y,z,L,n₀ from the array F_(rec) thus formed.

9.16. Calculating w_(p) and adjusting the focus

This step is carried out as described in 7.15. It can however be avoidedif the index of the sample is close to the nominal index of theobjectives, in which case it is possible to choose for examplew_(p)=L/2.

The arrays M_(k,p,q)[i,j] obtained with the procedure 9.9. used withoutvariant are substituted for those previously obtained with the procedure7.12.

Since the sampling is regular, we use K₀=K₁=K

The values imax_(k),jmax_(k) are now given by:${{imax}_{k} = {\frac{N_{d}}{N_{pix}}{{Io}\lbrack k\rbrack}}},{{jmax}_{k} = {\frac{N_{d}}{N_{pix}}{{{Jo}\lbrack k\rbrack}.}}}$

Step 2 of the procedure (4002) in FIG. 57 requires the acquisition ofthe value of the wave at the point (imax_(k),jmax_(k)). As the glassesare used to stop the direct illuminating wave, this value is notavailable. This step is thus replaced by Steps 2.1 and 2.2. below:

Step 2.1. For each value of k, the program calculates:$R_{k,0,0} = \frac{\sum\limits_{{({i,j})} \in E_{k}}\frac{{H_{0,0,0}\left\lbrack {i,j} \right\rbrack}\overset{\_}{H_{k,0,0}\left\lbrack {i,j} \right\rbrack}}{\left( {B_{0,0,0}\left\lbrack {i,j} \right\rbrack} \right)^{2} + \left( {B_{k,0,0}\left\lbrack {i,j} \right\rbrack} \right)^{2}}}{\sum\limits_{{({i,j})} \in E_{k}}\frac{{{H_{k,0,0}\left\lbrack {i,j} \right\rbrack}}^{2}}{\left( {B_{0,0,0}\left\lbrack {i,j} \right\rbrack} \right)^{2} + \left( {B_{k,0,0}\left\lbrack {i,j} \right\rbrack} \right)^{2}}}$

 where:$E_{k} = \left\{ \left( {i,j} \right) \middle| {\frac{{{H_{0,0,0}\left\lbrack {i,j} \right\rbrack}\overset{\_}{H_{k,0,0}\left\lbrack {i,j} \right\rbrack}}}{\left( {B_{0,0,0}\left\lbrack {i,j} \right\rbrack} \right)^{2} + \left( {B_{k,0,0}\left\lbrack {i,j} \right\rbrack} \right)^{2}} \geq {{Coef} \cdot {\max\limits_{\underset{0 \leq b \leq {N_{pix} - 1}}{0 \leq a \leq {N_{pix} - 1}}}\left( \frac{{{H_{0,0,0}\left\lbrack {a,b} \right\rbrack}\overset{\_}{H_{k,0,0}\left\lbrack {a,b} \right\rbrack}}}{\left( {B_{0,0,0}\left\lbrack {a,b} \right\rbrack} \right)^{2} + \left( {B_{k,0,0}\left\lbrack {a,b} \right\rbrack} \right)^{2}} \right)}}} \right\}$

and where for example Coef=0.5

Step 2.2. For each value of k,i,j the program carries out:${{Ms}_{k}\left\lbrack {i,j} \right\rbrack} = \frac{{{Ms}_{k}\left\lbrack {i,j} \right\rbrack}R_{k,0,0}}{{{Ra}\left\lbrack {0,{{Io}\lbrack k\rbrack},{{Jo}\lbrack k\rbrack}} \right\rbrack}{F_{rec}\left\lbrack {{{Io}\lbrack k\rbrack},{{Jo}\lbrack k\rbrack}} \right\rbrack}}$

where F_(rec) is the array determined in 9.15.

9.17. Obtaining the aberration compensation function

This step is carried out as described in 7.16. with K₀=K₁=K

9.18. Obtaining three-dimensional images by the method described in thethird embodiment

9.18.1. Without eliminating the direct wave

This step is carried out as described in 7.17.2.

the arrays M_(k,p,q)[i,j] H_(k,p,q)[i,j] B_(k,p,q)[i,j] BH_(k,p,q)[i,j]obtained with the procedure 9.9 used with Variant 2 are substituted forthose previously obtained with the procedure 7.12

because the sampling is regular, we use K₀=K₁=K, a₀=a₁=1.

the values imax_(k,p,q),jmax_(k,p,q) are now given by:imax_(k,p,q)=Io[k],jmax_(k,p,q)=Jo[k].

9.18.2. Eliminating the direct wave

The principle is the same as above but the procedure 9.9. is usedwithout a variant. In addition, Step 2 of the algorithm described in7.17.2. requires the acquisition of the value of the wave at the point(imax_(k,p,0),jmax_(k,p,0)) . As the glasses are used to stop the directilluminating wave, this value is not available. This step is thusreplaced by Step 2 below, equivalent to the one indicated in 7.18.1.

Step 2: For each value of k,p,q,i,j, the program performs:${M_{k,p,q}\left\lbrack {i,j} \right\rbrack} = \frac{{M_{k,p,q}\left\lbrack {i,j} \right\rbrack}{D_{{p\overset{\_}{q}} + {\overset{\_}{p}q}}\left\lbrack {i,j} \right\rbrack}R_{k,p,q}}{{{Ra}\left\lbrack {p,{{Io}\lbrack k\rbrack},{{Jo}\lbrack k\rbrack}} \right\rbrack}{F_{rec}\left\lbrack {{{Io}\lbrack k\rbrack},{{Jo}\lbrack k\rbrack}} \right\rbrack}{D_{p}\left\lbrack {{{Io}\lbrack k\rbrack},{{Jo}\lbrack k\rbrack}} \right\rbrack}}$

9.19. Obtaining three-dimensional images using a fast method with theelimination of direct illumination

The use of the previously described method has the drawback of requiringthe acquisition of reference images. With each reference imageacquisition, it is necessary to move the mirror (5113) considerably. Itis possible to use a reference image for the phase correction of severalsuccessive useful images, and hence not acquire a reference image witheach elementary image acquisition. Nevertheless, if the system is notperfectly stable over a time scale comparable to the acquisition timefor a complete three-dimensional frequency representation, numerousreference image acquisitions will be required. Owing to the significantdisplacement of the mirror that they require, these acquisitionsconstitute a great loss of time. To obviate the acquisition of referenceimages, Steps 1 and 2 of Paragraph 7.17.2., whose purpose is to allowthe phase correction of the two-dimensional frequency representations,can be replaced by the method described below. This method can also beused with the other embodiments, but is then of only limited interest.

This method includes a preliminary phase, which is carried out beforeany calculation requiring phase correction (i.e. before the actualimaging phase), then a modification of the steps used in the imagingphase. The acquisition of an image to be dealt with using this methodcan take place according to the procedure 9.9. used with Variant 1, i.e.without reference image acquisition and with the use of the glass tocancel the direct illumination.

9.19.1. Preliminary phase

The preliminary phase consists in determining, from a limited number oftwodimensional frequency representations indexed by the index l, thearrays Kn_(p,q)[ni,nj,l] Fh_(p,q)[ni,nj,l]. In these arrays:

p indexes the side (left or right) towards which the direct illuminatingwave goes

q indexes the side from which the frequency representation comes(opposite or not opposite the side on which the direct illuminating wavearrives)

l indexes the position of the point corresponding to the directilluminating wave, on the image generated by the method described in9.9., from the side indexed by q=0.

The indices (l,p) thus characterize an illuminating wave and the indices(l,p,q) characterize a two-dimensional frequency representationcorresponding to this illuminating wave and to the sensor indexed by q.

The indices ni,nj are the coordinates on two axes of this frequencyrepresentation, after centering in relation to the point of directimpact of the illuminating wave.

The array Kn_(p,q)[ni,nj,l] contains the third coordinate nk of theconsidered frequency representation, for each pair ni, nj.

The array Fh_(p,q)[ni,nj,l] contains the value of this representation atthe point of coordinates ni,nj,nk after phase correction in relation tothe representation defined by l=0.

The array Bh_(p,q)[ni,ni,l] contains the noise on the correspondingelements of Fh_(p,q)[ni,nj,l].

These arrays thus characterize the values of the three-dimensionalfrequency representation of the object at a certain number of points.The array Kn_(p,q) characterizes the points at which values areavailable, and the array Fh_(p,q) characterizes these values themselves.The positions of the direct illuminating wave corresponding to theindices l are chosen so that every two-dimensional frequencyrepresentation, after recentering, has a non-empty intersection with thepart of the three-dimensional frequency representation of the objectconstituted by the superposition of the representations corresponding tothe different indices l, and characterized by the arrays Kn_(p,q) andFh_(p,q).

The preliminary phase is broken down into three steps:

Step 1: FIG. 81 represents the points corresponding to the directilluminating wave, on the image generated by procedure 9.9. from sensorsreached by this direct illuminating wave, for several values of l. Wedenote IT[l],JL[l] the coordinates of point indexed by l, with forexample:

1 IT[l] JL[l] 0 $\frac{N_{pix}}{2}$

$\frac{N_{pix}}{2}$

1 N_(pix) − 1 − marge $\frac{N_{pix}}{2}$

2 $\frac{N_{pix}}{2}$

N_(pix) − 1 − marge 3 marge $\frac{N_{pix}}{2}$

4 $\frac{N_{pix}}{2}$

marge

where marge=10 for example.

The points indexed by l must form part of the path defined by the arraysIo and Jo. The program then determines, for each value of l, the valueof k such that IL[l]=Io[k] and JL[l]=Jo[k]. It stores this value thearray TK at TK[l].

Step 2: The second step consists in determining the arrays Kn_(p,q) andFh_(p,q)

The program first initializes these arrays, for example to a value of−10000.

The program then goes through all the quintuplets (l,p,q,i,j). Itcalculates for each of them

k=TK[l]

ni=i−imax_(k,p,q) +N _(pix)

nj=j−jmax_(k,p,q) +N _(pix)

${nk} = {\sqrt{\left( {\frac{n_{o}}{n_{v}}\quad K} \right)^{2} - \left( {i - \frac{N_{pix}}{2}} \right)^{2} - \left( {j - \frac{N_{pix}}{2}} \right)^{2}} - \sqrt{\left( {\frac{n_{o}}{n_{v}}\quad K} \right)^{2} - \left( {{imax}_{k,p,q} - \frac{N_{pix}}{2}} \right)^{2} - \left( {{jmax}_{k,p,q} - \frac{N_{pix}}{2}} \right)^{2}} + N_{pix}}$

It takes for each of these numbers the nearest integer and thenperforms:

Kn _(p,q) [ni,nj,l]=nk

Fh _(p,q) [ni,nj,l]=M _(k,p,q) [i,j]D_(p{overscore (q)}+{overscore (p)}q) [i,j]

Bh _(p,q) [ni,nj,l]=B _(k,p,q) [i,j]|D_(p{overscore (q)}+{overscore (p)}q) [i,j]|

Step 3: This step consists in modifying the array Fh_(p,q). The programgoes through all the (l,p,q). For each value of this triplet, theprogram carries out Operations 1 to 3 below:

Operation 1: The program initializes to 0 the numbers nom and denom

Operation 2: The program goes through all the values of (ni,nj) testingthe condition |Kn_(p,q)[ni,nj,l]−Kn_(p,q)[ni,nj,0]|²≦lim with, forexample, lim=1. When this condition is met, it performs:${nom}_{p,q,l}+=\frac{{{Fh}_{p,q}\left\lbrack {{ni},{nj},0} \right\rbrack}\quad \overset{\_}{{Fh}_{p,q}\left\lbrack {{ni},{nj},l} \right\rbrack}}{\left( {{Bh}_{p,q}\left\lbrack {{ni},{nj},0} \right\rbrack} \right)^{2} + \left( {{Bh}_{p,q}\left\lbrack {{ni},{nj},l} \right\rbrack} \right)^{2}}$${denom}_{p,q,l}+=\frac{{{{Fh}_{p,q}\left\lbrack {{ni},{nj},l} \right\rbrack}}^{2}}{\left( {{Bh}_{p,q}\left\lbrack {{ni},{nj},0} \right\rbrack} \right)^{2} + \left( {{Bh}_{p,q}\left\lbrack {{ni},{nj},l} \right\rbrack} \right)^{2}}$

Operation 3: The program performs:${{Fh}_{p,q}\left\lbrack {{ni},{nj},l} \right\rbrack} = {\frac{{nom}_{p,q,l}}{{denom}_{p,q,l}}\quad {{Fh}_{p,q}\left\lbrack {{ni},{nj},l} \right\rbrack}}$

9.19.2. Imaging phase

The imaging phase differs from the one used in 7.17.2. by the phasecorrection method. Phase correction here is carried out in relation tothe part of the frequency representation of the object characterized bythe arrays calculated in the preliminary phase, and not in relation tothe illuminating wave image point, to prerecorded values of theilluminating wave or to reference images.

Steps 1, 2 and 3 below replace Steps 1 and 2 defined in 7.17.2.

Step 1: The program carries out, for all the values of k,p,q,i,j:

M _(k,p,q) [i,j]=M _(k,p,q) [i,j]D _(p{overscore (q)}+{overscore (p)}q)[i,j]

 B _(k,p,q) [i,j]=B _(k,p,q) [i,j]|D_(p{overscore (q)}+{overscore (p)}q) [i,j]|

Step 2: This step consists in establishing the complex coefficientcharacterizing, for each two-dimensional representation, the phase andintensity shift between this two-dimensional representation and theportion of three-dimensional representation characterized by the arraysKn_(p,q) and Fh_(p,q.) This complex coefficient, for the representationcharacterized by the indices k,p,q, is expressed in the form$\frac{{nom}_{k,p,q}}{{denom}_{k,p,q}}.$

It is obtained by carrying out a loop on all the indices (k,p,q,i,j,l).For each (k,p,q,i,j,l):

The program calculates:

ni=i−imax_(k,p,q) +N _(pix)

nj=j−jmax_(k,p,q) +N _(pix)

${nk} = {\sqrt{\left( {\frac{n_{o}}{n_{v}}K} \right)^{2} - \left( {i - \frac{N_{pix}}{2}} \right)^{2} - \left( {j - \frac{N_{pix}}{2}} \right)^{2}} - \sqrt{\left( {\frac{n_{o}}{n_{v}}K} \right)^{2} - \left( {{imax}_{k,p,q} - \frac{N_{pix}}{2}} \right)^{2} - \left( {{jmax}_{k,p,q} - \frac{N_{pix}}{2}} \right)^{2}} + N_{pix}}$

It tests the condition:

|Kn _(p,g) [ni,nj,l]−nk| ²≦lim with, for example, lim=1.

If the condition is true, the program performs the operations${{nom}_{k,p,q}\text{+}} = \frac{\overset{\_}{M_{k,p,q}\left\lbrack {i,j} \right\rbrack}{{Fh}_{p,q}\left\lbrack {{ni},{nj},l} \right\rbrack}}{\left( {B_{k,p,q}\left\lbrack {i,j} \right\rbrack} \right)^{2} + \left( {{Bh}_{p,q}\left\lbrack {{ni},{nj},l} \right\rbrack} \right)^{2}}$${{denom}_{k,p,q}\text{+}} = \frac{\left| {M_{k,p,q}\left\lbrack {i,j} \right\rbrack} \right|^{2}}{\left( {B_{k,p,q}\left\lbrack {i,j} \right\rbrack} \right)^{2} + \left( {{Bh}_{p,q}\left\lbrack {{ni},{nj},l} \right\rbrack} \right)^{2}}$

Step 3: This step constitutes the phase correction proper. The programperforms:${M_{k,p,q}\left\lbrack {i,j} \right\rbrack} = {\frac{{nom}_{k,p,q}}{{denom}_{k,p,q}}{M_{k,p,q}\left\lbrack {i,j} \right\rbrack}}$

9.20. Obtaining threedimensional images with a simplified method

To generate the three-dimensional image of the object, we can confineourselves to the representation F_(0,0) defined in 7.17. This isequivalent, in the procedure described in 7.17.2., to adopting arraysIB_(p,q) which are zero for any pair (p,q)≠(0,0).

It is also assumed here that the object has an average refractive indexclose to the nominal index of the object observed and that the opticaltable is totally devoid of vibrations.

Steps 9.11. to 9.17 can then be eliminated. The present method differsmoreover from the preceding one in the method used to adjust theposition of the objectives before utilization and in the imagesuperposition algorithm.

9.20.1. Adjustment of objectives and of mirror (5113)

This adjustment can be carried out with a transparent plate provided theobjectives are not moved after adjustment, when the object isintroduced. If objectives designed to operate without any immersionliquid or any cover glass are involved (nominal index equal to 1) or ifthe sample is not very thick or has an average index close to 1, it canalso be carried out without an object. During this step, the beams FEGand FRD are used.

The mirror (5113) must be removable and must be replaceable by a plateabsorbent at every point except at a central point where a reflector ofsmall dimensions is placed. The dimensions of this reflector should beabout $\frac{D}{2N_{pix}}$

where D is the diameter of the beam incident on the mirror (5113). Thisplate must be temporarily placed on the mirror so that the reflectoroccupies roughly the center of the beam incident on the mirror. Themirror positioner must itself be fixed on a three-axis translationpositioner.

This adjustment must be carried out immediately after the series ofadjustments described in 9.5., and the objectives must not be moved anymore thereafter. It includes the following steps:

Step 1: The mirror is replaced by the absorbing plate

Step 2: Using only the beam FRD, the mirror is translated so that theimage produced on the CCD (4339) is a centered point image.

Step 3: Using only the beam FEG, the objectives are moved so that theimage produced on the CCD (4339) is a centered point image.

Step 4: The mirror can then be put back in place.

9.20.2. Algorithm for calculating the three-dimensional representation

Steps 1 and 2 of the algorithm described in 7.17.2. can be eliminated.In fact, the additional adjustment carried out and the absence ofvibrations make it possible to avoid any phase shift in the illuminatingbeam.

9.21. Use of objectives exhibiting aberrations

As in the third embodiment, objectives of the type described in 7.20 or7.21 can be used. The corresponding modifications of the algorithms aresimilar to those described in 7.20 and 7.21. It is of course necessaryto adapt them to the method for obtaining the plane frequency image,which differs significantly from the third embodiment.

10. OPTICAL ELEMENT POSITIONING DEVICE

The embodiments described, and in particular Embodiment 4, require theuse of many high-precision positioners. These positioners are costlyelements ill suited to mass production and capable of losing theiradjustment with time. These positioners, with the exception of thepositioners of the objectives and the sample, should in principle beadjusted only once, during the initial adjustment phase.

A solution to this problem is to use removable positioners during themanufacture of the microscope. After positioning, each element can befixed with an appropriate adhesive. For example, in Embodiment 4, theSLMs may be secured by means of the device of FIGS. 83 to 85. The partof the attachment device which is incorporated in the microscopecomprises three mobile assemblies:

Assembly 1: Consisting of the following elements, connected together:

an attachment plate (5801) shown in detail in FIG. 83 and having aporous surface for gluing (5802).

the SLM (5804) fixed on the unglued part (5803) of this plate.

a plate (5808) in magnetizable material, for example iron, fixed to therear of the plate (5801).

Assembly 2: Consisting of the following elements, connected together:

a plate (5805) shown in detail in FIG. 84 exhibiting a porous surfacefor gluing (5806) and a hollow (5807) at its center.

a plate (5809) exhibiting a surface for gluing (5817).

a plate (5810) in magnetizable material, for example iron, fixed to theplate (5809).

Assembly 3: Consisting of a plate (5811) exhibiting a porous surface forgluing (5816).

The removable part of the attachment device includes the followingassemblies:

Assembly 4: Consisting of an arm (5815) with a magnetizable part (5814)and connected to a positioner POS1 not shown. The part (5814) comprisesan electric wire winding around an iron core, not shown. By supplyingthis winding, (5814) and (5810) are brought together and, eliminatingthe power supply, these elements are disconnected.

Assembly 5: Consisting of an arm (5813) with a magnetizable part (5812)and connected to a positioner POS2 not shown. The part (5812) comprisesan electric wire winding around an iron core, not shown. By supplyingthis winding, (5808) and (5812) are brought together and, eliminatingthe power supply, these elements are disconnected.

The fixed part of positioner POS1 and the element 3 are integral withthe optical table. The “fixed” part of positioner POS2 is integral withthe mobile part of positioner POS1. Positioner POS1 allows single-axisdisplacement in the direction of the axis {right arrow over (i)} and arotation around the axis {right arrow over (j)}. Positioner POS2 allowstranslation along each of the vectors {right arrow over (j)} and {rightarrow over (k)} and rotation around the axis {right arrow over (i)}. Italso allows, but with a very small adjustment margin, rotation aroundthe axis {right arrow over (k)}.

For the positioning of the system, the above-designated surfaces to beglued are first glued. The magnets of the parts (5812) and (5814) arepowered so as to connect the removable parts and the non-removableparts. The adjustment is carried out normally with positioners POS1 andPOS2. The system is left in place sufficiently long so that the gluedries. The magnets are then no longer powered so as to detach theremovable parts from the non-removable parts. The adjustment is thenfinal and the removable part can be removed.

The glue used must have a sufficiently long setting time so as not tohinder adjustment and must have a minimum shrinkage when it dries. It isalso possible to provide specific orifices in the plates (5811) and(5805) to inject glue after positioning. This example is given for thepositioning of the SLMs but is adaptable simply to all the elements tobe positioned in the system. Depending on the number of degrees offreedom required and the type of element to be positioned, the form ofthe mobile platforms must be adapted. The principle consisting in usingremovable positioners and in carrying out final attachment by gluinghowever remains valid.

11. SUPPORT SUITED TO THE TRANSPORT AND MAINTAINING OF ADJUSTMENTS MADE

The microscopes described in Embodiments 3 to 5 are made up of a numberof elements fixed to an optical table. During possible transport, evenlight shocks can lead to a disadjustment of the system. During prolongeduse, dust may become deposited on the different optical elements.

To deal with these problems, the microscope described can be protectedby a system whose principle is shown in FIG. 86. The optical table,which can be in granite, for example, constitutes the lower part of ahermetically closed box (5901). The fact that the box (5901) ishermetically closed protects the assembly from dust. The box (5901) isincluded in a larger box (5902), without any direct contact between thetwo boxes. The two boxes are separated by shock absorbers which may berubber balloons (5903) inflated in an appropriate manner, and which areplaced on the six sides of the box (5901). This system allows theabsorption of shocks and prevents loss of adjustment during transport,while guaranteeing good suspension of the optical table duringutilization.

However, the part of the system made up of the two objectives and theirpositioners must remain accessible. This leads to certain adaptations inthe form of the boxes, shown in FIG. 87, adapted here to the exampleconstituted by the embodiments 4 and 5. The front wall of the box (5901)constitutes a vertical plane passing in FIG. 63 between the mirrors(4451) and (4452). The box (5901) has a projection (5903) allowing theattachment of the mirrors (4454)(4455)(4456) and of the objective (4317)under the plane of the optical table proper. The box (5902) shown inbroken lines has a recess (5904) accommodating the objectives and thesample.

In order for the mirrors (4453) and (4454), as well as the inlets of theobjectives, to remain inaccessible, and in order to prevent any ingressof dust, the form of the box (5902) must also be adapted locally. Thisadaptation is shown in detail in FIG. 88. The box has two projections(5905) and (5906) containing respectively the mirrors (4453) and (4454),and has two openings linked to the inlet of the objectives (in fact, tothe mounts of these objectives) by rubber sleeves (6001) (6002). Thepositioners of the objectives and of the sample, not shown, are outsidethe box (5902).

In FIG. 87, the shock absorbers (5903) are not shown, but they arepresent in the entire zone between the two boxes.

A cover protecting the accessible part (objectives and sample, as wellas their positioners) must also be provided.

In the case of Embodiment 5, the outside box (5901) must include anadditional compartment containing the elements which are not on theoptical table.

12. VARIANTS

Other embodiments are of course possible and the description above isnot limitative. It is in particular possible to use more objectives, orto have other combinations of the types of beam deflector or receiver,or modify the phase correction algorithms.

13. INDUSTRIAL APPLICATION POSSIBILITIES 13.1. REFERENCES

[Thomas]: 4-D imaging software observe living cells, Charles Thomas &John White, Scientific Computing World p.31, décembre 1996.

[Holton]: Under a Microscope: Confocal Microscopy Casts New Light on theDynamics of Life, W. Conard Holton, Photonics Spectra p.78, février1995.

[Pike]: Phase measuring scanning optical microscope, J. G. Walker & E.R. Pike, brevet WO 91/07682

[Bertero]: Analytic inversion formula for confocal scanning microscopy,B. Bertero, C.Of Mol, E. R. Pike, Journal of the Optical Society ofAmerica vol.4 no.9, septembre 1987

[Ueki]: Three-dimensional optical memory with a photorefractive crystal,Y. Kawata, H. Ueki, Y. Hashimoto, S. Kawata, Applied Optics vol.34 no 20p.4105, 10 juillet 1995

[Juskaitis]: Differential phase-contrast microscope with a splitdetector for the readout system of a multilayered optical memory, Y.Kawata, R. Juskaitis, T. Tanaka, T. Wilson & S. Kawata, Applied Opticsvol.35 no 14 p.2466, 10 mai 1996

[Parthenopoulos]: Three-dimensional optical storage memory, D. A.Parthenopoulos & P. M. Rentzepis, Science 245, p.843, 1989

[Strickler]: Three dimensional optical data storage in refractive mediaby two-photon excitation, J. H. Strickler & W. W. Webb, Optics Letters16, p.1780, 1991

[McMichael]: Compact holographic storage demonstrator with rapid access,I. McMichael, W. Christian, D. Pletcher, T. Y. Chang & J. H. Hong,Applied Optics vol.35 no 14 p.2375, 10 mai 1996.

[Bashaw]: Cross-talk considerations for angular and phase-encodedmultiplexing in volume holography, M. C. Bashaw, J. F. Heanue, A.Aharoni, J. F. Walkup & L. Hesselink, Journal of the Optical Society ofAmerica B vol.11 no 9 p.1820 septembre 1994

[Barbarstatis]: Shift multiplexing with spherical reference waves, G.Barbarstatis, M. Levene, D. Psaltis, Applied Optics vol.35 no 14 p.2403,10 mai 1996

13.2. Discussion

Prior-art microscopes form, by an optical process, a two-dimensionalimage corresponding to a magnified section of the observed object. Thisimage may, if desired, be recorded by a video camera for subsequentrestitution.

It is possible to generate a three-dimensional image using one of thesemicroscopes and varying the focussing adjustment. To each adjustmentthere corresponds a different section plane, and an image in threedimensions can be reconstituted from these section planes. Somemicroscopes equipped with a powered focussing device and appropriatesoftware perform this operation automatically. Such microscopes aredescribed for example in [Thomas]. The major drawback of thesemicroscopes is that the image of a section plane is highly disturbed bythe content of the other planes.

There are also confocal microscopes in which the illumination isfocussed on a point and the three-dimensional image is generated byscanning all the points of the object. Such microscopes are describedfor example in [Holton]. These microscopes make it possible to solve theproblem of the systems described in [Thomas], namely that the valuedetected at a given point is not greatly disturbed by the value ofneighboring points.

Confocal microscopes have the drawback of being able to detect only theintensity of the received wave and not its phase. As many objectscurrently observed are characterized essentially by variations inrefractive index leading to variations in the phase of the transmittedwave, this drawback is significantly troublesome for users who mustcolor the images observed. For this reason, efforts were made to designconfocal microscopes sensitive to the phase of the transmitted wave[Pike]. For various reasons, these microscopes are still not veryeffective.

The image generated by confocal microscopes does not have the besttheoretical definition obtainable in principle from the wave received bythe objective used. This is related, as indicated in 7.18.3., to thefact that the confocal microscope method filters the high frequenciessignificantly. For this reason, efforts have been made to improve theresolution of these confocal microscopes [Bertero].

The present microscope remedies these shortcomings of prior-artmicroscopes in terms of resolution and in terms of phase detection. Itcan be used advantageously as a replacement for such microscopes, in alltheir applications.

A new field of application for microscopes is the reading ofthree-dimensional optical memories.

A first type of optical memory is the one in which data are recordedpoint by point in a three-dimensional material in the form of variationsin the local properties of this material ([Ueki], [Parthenopoulos],[Strickler]). These data must thus be read by a microscope capable ofreading three-dimensional data without any mutual disturbance betweenthe data recorded in several successive layers of material. Thedisturbance of the image of a point by rays diffracted by neighboringpoints thus results here in inter-symbol interference. In general, theauthors have used few layers at great distances from each other, thuslimiting these interferences. However, if a greater quantity of datamust be recorded in a given volume, conventional microscopy methods areinadequate. In particular in the case of [Strickler] and [Ueki], dataare recorded in the form of index variations and the confocal microscopeis particularly ill suited to their reading. For this reason, effortshave been made to improve the data reading system [Juskaitis].

The present microscope constitutes the reading solution allowing maximumintegration of this type of memory. In fact, the image that it makesobtainable takes into account the index and reduces inter-symbolinterference significantly. In the ideal case in which the entire beamcoming from the sample is detected, which may be accomplished byincreasing the number or the aperture of the objectives, inter-symbolinterference is entirely eliminated.

Another type of optical memory is the holographic memory. For example,in the document [McMichael], data are read by illuminating the objectconstituted by the optical memory with a parallel beam of variabledirection and by detecting the wave coming from the sample for eachdirection of the illuminating beam. A direction of the illuminating beamcorresponds to a page of data and each point of the two-dimensionalfrequency representation of the wave coming from the object for a givenilluminating wave corresponds to a bit stored in the optical memory. Aseach point of the two-dimensional representation of the wave coming fromthe object moreover corresponds to a point of the three-dimensionalrepresentation of the object itself, a bit stored in the optical memorycorresponds to a point of the three-dimensional frequency representationof this optical memory. An analysis of this type of memory in terms offrequency representations may be found in [Bashaw].

The present microscope can thus be used advantageously to read suchoptical memories, the bits stored en memory corresponding directly topoints of the three-dimensional frequency representation obtained by thepresent microscope from the object constituted by the optical memory.The writing system of the optical memory must however be designed so asnot to use the points of the three-dimensional frequency representationwhich are not obtained by the present microscope, except to increase thenumber of objectives used in order to avoid non-detection of certainfrequencies.

Other types of holographic optical memories exist [Barbarstatis]. Ingeneral, the present microscope makes it possible to obtain arepresentation of the observed object which would be “perfect” in theideal case in which the objectives used would cover the entire spacearound the object. In the case in which the representation obtained isperfect, the data stored in memory and detected in the form of athree-dimensional frequency representation of the object can then berestored in any form: it is possible to simulate, by using the knownrepresentation of the object constituted by the optical memory, the wavewhich would be obtained from any illumination or by any other readingmethod (at the wavelength used by the microscope). All types of opticalmemory may thus be read by the present microscope, in the general caseby means of additional operations allowing the reconstitution of thedata from the frequency representation of the object constituted by theoptical memory. In the case in which the representation is not perfect,adequate precautions must be taken in order to take into account the“shadow zones” of the frequency representation of the object.

What is claimed is:
 1. A microscope comprising: a) means providing alaser beam, b) means for splitting the laser beam into a reference waveand an illuminating wave, c) optical means for rendering theilluminating wave plane and for directing it towards an observed object,the illuminating wave being diffracted by the observed object to producea diffracted wave, d) optical means for superimposing the diffractedwave and the reference wave, to produce an interference pattern, e) atleast one receiver for receiving and recording the interference pattern,f) means for calculating, from the recorded interference pattern, afrequency representation of the diffracted wave, corresponding to phasesand amplitudes of plane wave components of the diffracted wave, g) abeam deflector for varying the direction of the illuminating wave from afirst direction to a second direction, for obtaining a first frequencyrepresentation of the diffracted wave corresponding to the firstdirection of the illuminating wave, and a second frequencyrepresentation of the diffracted wave corresponding to the seconddirection of the illuminating wave, h) means for applying a first phaseshift to the first frequency representation, i) means for superimposingthe first frequency representation and the second frequencyrepresentation to obtain a three-dimensional frequency representation ofthe observed object, wherein the first phase shift is such that the partof the three-dimensional frequency representation corresponding to thefirst frequency representation has substantially the same phasereference as the part of the three-dimensional frequency representationcorresponding to the second frequency representation.
 2. The microscopeof claim 1, wherein the means for applying the first phase shift to thefirst frequency representation are calculation means.
 3. The microscopeof claim 2, wherein said calculation means multiply the first frequencyrepresentation by a complex number having a phase corresponding to thefirst phase shift.
 4. The microscope of claim 2, wherein said beamdeflector varies the direction of the illuminating wave from the seconddirection to a third direction and to a fourth direction, for obtaininga third frequency representation of the diffracted wave corresponding tothe third direction of the illuminating wave, and a fourth frequencyrepresentation of the diffracted wave corresponding to the fourthdirection of the illuminating wave, and further comprising means forsuperimposing the first frequency representation and the third frequencyrepresentation to obtain a first sub-representation of the observedobject, and means for superimposing the second frequency representationand the fourth frequency representation to obtain a secondsub-representation of the observed object, and wherein said means forapplying a first phase shift comprise means for applying a second phaseshift to the first sub-representation of the observed object, thusshifting the phase of the first frequency representation included in thefirst sub-representation, and further comprising means for superimposingthe first sub-representation and the second sub-representation to obtainthe three-dimensional frequency representation of the object, thussuperimposing the first frequency representation included in the firstsub-representation and the second frequency representation included inthe second sub-representation.
 5. The microscope of claim 1, furthercomprising means for causing the phase difference between theilluminating wave and the reference wave to be reproducible.
 6. Themicroscope of claim 5, wherein said means for applying a first phaseshift comprise an optoelectronic device for shifting the phase of theilluminating and/or the reference wave, thus shifting the phase of thefirst frequency representation.
 7. The microscope of claim 5, whereinsaid means for applying a first phase shift comprise adjustment meansfor adjusting the position of an optical element, so as tosimultaneously adjust the phase shifts applied to the illuminating wavefor a plurality of directions of the illuminating wave.
 8. Themicroscope of claim 1, further comprising means for determining thefirst phase shift.
 9. The microscope of claim 8, wherein said means fordetermining the first phase shift determine the phase difference betweenat least a first point of the second frequency representation and acorresponding second point of the first frequency representation, saidfirst and second points being superimposed in the three-dimensionalfrequency representation.
 10. The microscope of claim 9, wherein saidmeans for determining the first phase shift calculate a complex numberhaving a phase which is an evaluation of said phase difference.
 11. Themicroscope of claim 10, wherein said means for determining the firstphase shift calculate a weighted average, over a plurality of the firstpoints and the corresponding second points, of the product of thecomplex value of the second frequency representation at the second pointby the conjugate of the complex value of the first frequencyrepresentation at the first point, for obtaining said complex number.12. The microscope of claim 1, further comprising means for applying asecond phase shift to the second frequency representation of thediffracted wave.
 13. The microscope of claim 12, wherein said means forapplying the first and second phase shifts consist of means for dividingthe first frequency representation by the value of the first frequencyrepresentation at a first point, and means for dividing the secondfrequency representation by the value of the second frequencyrepresentation at a second point, wherein the first and second pointsare superimposed to the point of origin in the three-dimensionalfrequency representation.
 14. The microscope of claim 1, wherein theobserved object, and said at least one receiver, are fixed in position.15. The microscope of claim 1, further comprising means for multiplyingthe first and second frequency representations by a set of complexvalues for cancelling spherical aberration.
 16. The microscope of claim1, further comprising: a) a first microscope objective lens disposedalong the path of the illuminating wave directed towards the observedobject, in the first direction of the illuminating waves b) a secondmicroscope objective lens disposed along the path of the illuminatingwave directed towards the observed object, in the second direction ofthe illuminating wave, c) said at least one receiver comprising a firstreceiver unit for recording the wave coming from the observed objectafter passing through the first objective, and at least a secondreceiver unit for recording the wave coming from the observed objectafter passing through the second objective.
 17. The microscope of claim1, further comprising a) means for polarizing the illuminating wave intwo directions orthogonal to each other, and b) means for analyzing thediffracted wave in two directions orthogonal to each other.
 18. Themicroscope of claim 1, wherein the beam deflector comprises: a) aprimary beam deflector for generating small variations of the directionof a parallel beam, and b) a condenser lens for transforming the smallvariations into large variations of the direction of the illuminatingbeam.
 19. The microscope of claim 18, wherein said primary beamdeflector comprises a first spatial modulator controlled so as togenerate an exiting beam from an incident parallel beam of givendirection, the direction of the exiting beam being controlled by thestate of the modulator.
 20. The microscope of claim 19, wherein saidspatial modulator is a binary modulator.
 21. The microscope of claim 20,further comprising: a) a first intermediate lens located along the pathof the exiting beam, b) a diaphragm and/or a second spatial modulatorplaced in the image focal plane of the first intermediate lens toeliminate spurious beams generated by the first spatial modulator, andc) a second intermediate lens located along the path of the exiting beamdownstream of said diaphragm to convert the beam exiting from thediaphragm into a parallel beam.
 22. In a microscope, a method forgenerating a three-dimensional frequency representation of an observedobject, comprising the steps of: a) splitting a laser beam into areference wave and an illuminating wave, b) rendering the illuminatingwave plane c) directing the plane iluminating wave in a first directionfor illuminating the observed object, the illuminating wave beingdiffracted by the object to provide a diffracted wave, d) superimposingthe diffracted wave and the reference wave to produce an interferencepattern, e) recording the interference pattern, f) calculating from atleast the interference pattern a first frequency representation of thediffracted wave corresponding to phases and amplitudes of plane wavecomponents of the diffracted wave, g) changing the direction of theilluminating wave and repeating steps (d) to (f) at least once, toproduce at least a second frequency representation of the diffractedwave, h) applying a first phase shift to the first frequencyrepresentation, and i) superimposing the first frequency representationand the second frequency representation, to obtain a three-dimensionalfrequency representation of the observed object, wherein the first phaseshift is such that the part of the three-dimensional frequencyrepresentation corresponding to the first frequency representation hassubstantially the same phase reference as the part of saidthree-dimensional frequency representation corresponding to the secondfrequency representation.
 23. The method according to claim 22, whereinsaid step of applying a first phase shift comprises multiplying thefirst frequency representation by a complex number having a phase whichis equal to said first phase shift.
 24. The method according to claim22, further comprising the steps of: a) varying the direction of theilluminating wave from the second direction to a third direction and toa fourth direction for obtaining a third frequency representation of thediffracted wave corresponding to the third direction of the illuminatingwave and a fourth frequency representation of the diffracted wavecorresponding to the fourth direction of the illuminating wave b)superimposing the first frequency representation and the third frequencyrepresentation to obtain a first sub-representation of the observedobject, and superimposing the second frequency representation and thefourth frequency representation to obtain a second sub-representation ofthe observed object, c) applying the first phase shift to the firstsub-representation of the observed object, thus shifting the phase ofthe first frequency representation included in the firstsub-representation, d) superimposing the first frequency representationand the second frequency sub-representation to obtain thethree-dimensional frequency representation of the object, thussuperimposing the first frequency representation included in the firstsub-representation and the second frequency representation included inthe second sub-representation.
 25. The method according to claim 22,wherein said step of directing the illuminating wave in a firstdirection is performed such that the phase difference between theilluminating wave and the reference wave is reproducible.
 26. The methodaccording to claim 25, wherein the step of applying a first phase shiftis performed by shifting the phase of the reference wave and/or thephase of the illuminating wave when the illuminating wave is directed inthe first direction, thus shifting the phase of the first frequencyrepresentation.
 27. The method according to claim 26, wherein the shiftof the phase of the illuminating wave and/or the reference wave isperformed by applying independent phase shifts for each direction of theilluminating wave.
 28. The method according to claim 26, furtherproviding a step of adjusting the position of an optical element forsimultaneously adjusting the phase shifts applied to the illuminatingwave for a plurality of directions of the illuminating wave.
 29. Themethod according to claim 22, further comprising the step of determiningthe first phase shift.
 30. The method according to claim 29, whereinsaid step of determining the first phase shift comprises determining thephase difference between at least a first point of the second frequencyrepresentation and a corresponding second point of the first frequencyrepresentation to obtain the first phase shift, said first and secondpoints being superimposed in the three-dimensional frequencyrepresentation.
 31. The method according to claim 30, wherein said stepof determining the first phase shift consists of determining a complexnumber having a phase which is equal to said phase difference.
 32. Themethod according to claim 31, wherein said step of determining the firstphase consists of calculating a weighted average, over a plurality ofthe first points and of the corresponding second points, of the productof the complex value of said second frequency representation at saidsecond point, by the conjugate of the complex value of said firstfrequency representation at said first point.
 33. The method accordingto claim 22, further comprising the step of applying a second phaseshift to the second frequency representation.
 34. The method accordingto claim 33, wherein said step of applying a first phase shift consistsof dividing the first frequency representation by its value at a firstpoint of the first frequency representation and wherein said step ofapplying a second phase shift consists of dividing the second frequencyrepresentation by its value at a second point, said first and secondpoints being superimposed to the point of origin in thethree-dimensional frequency representation.
 35. The method according toclaim 22, further comprising the step of multiplying said first andsecond frequency representations by a set of complex values, to cancelthe spherical aberration.
 36. The method according to claim 22, whereinthe interference pattern is recorded on fixed receiver means and whereinthe observed object remains fixed in position.
 37. The method of claim22, further comprising the steps of: j) directing the illuminating wavethrough a first microscope objective towards said observed object, toobtain said first direction of the illuminating wave, k) recording theinterference pattern formed by the reference wave and by the wavediffracted by the object and having passed through the first microscopeobjective, l) recording the interference pattern formed by the referencewave and by the wave diffracted by the object and having passed througha second microscope objective, and m) directing the illuminating wavethrough the second microscope objective towards said observed object, toobtain said second direction of the illuminating wave, and repeatingsteps k) and l).
 38. The method of claim 22, further comprising thesteps of: a) polarizing the illuminating wave in two directionsorthogonal to each other, and b) analyzing the diffracted wave in twodirections orthogonal to each other.
 39. In a microscope, a method forimproving an initial three-dimensional frequency representation of anobserved object, comprising the steps of: a) splitting a laser beam intoa reference wave and an illuminating wave, b) rendering the illuminatingwave plane and directing it in a first direction for illuminating theobserved object, thereby providing a diffracted wave diffracted by theobject, c) superimposing the diffracted wave and the reference wave, toproduce an interference pattern, d) recording the interference pattern,e) calculating, from at least the interference pattern, a firstfrequency representation of the diffracted wave corresponding to phasesand amplitudes of plane wave components of the diffracted wave, f)applying a first phase shift to the first frequency representation, andg) superimposing the first frequency representation and the initialthree-dimensional frequency representation, to obtain a final frequencyrepresentation, wherein the first phase shift is such that the part ofthe three-dimensional frequency representation corresponding to thefirst frequency representation has substantially the same phasereference as the part of said three-dimensional frequency representationcorresponding to the second frequency representation.